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Detailed Modeling of the Volume Fraction Profile of Adsorbed Polymer Layers Using Small-Angle Neutron Scattering John C. Marshall, Terence Cosgrove,* Frans Leermakers,† Timothy M. Obey, and Ce´cile A. Dreiss School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom, and Laboratory for Physical and Colloid Chemistry, Agricultural University, Dreyenplein 6, 6703 H B Wageningen, The Netherlands Received October 7, 2003. In Final Form: February 26, 2004 A detailed analysis of on and off-contrast small-angle neutron scattering from poly(ethylene oxide)s adsorbed on polystyrene latex is presented. The results have been fitted to an exponential decay, selfconsistent mean-field (SCF) and scaling models of the volume fraction profile. As the chain length increases a clear self-similar layer emerges in both the SCF and scaling profiles. The RMS thickness of the adsorbed layer in the plateau of the adsorption isotherm varies as M0.4 for both the SCF and scaling profiles but the exponential one however gives a much lower exponent consistent with the neglect of the tail or distal region. The new constrained version of the Scheutjens-Fleer model was able to predict the structure of the layer and hence the scattering with great accuracy.
Introduction It is well established that small-angle neutron scattering (SANS) can be used to determine the structure of polymer layers adsorbed upon colloidal dispersions in a wide range of systems.1-15 The polymer volume fraction profile can be determined by two different SANS methods. The most straightforward technique employs isotopic substitution of the solvent to adjust the scattering length density of the solvent to match that of the substrate particles, this is commonly referred to as “contrast-matching”. In this case, the scattering originates only from the polymer layer. The second, and more complicated method, requires the dispersion of the coated and bare particles in a solvent with a scattering length density that is different to that of the particles and the adsorbed layer. These data can be manipulated to determine the scattering from the interference between the adsorbed layer and the colloidal particles.5,6,16 † Laboratory for Physical and Colloid Chemistry, Agricultural University.
(1) Crowley, T. L. Ph.D. Thesis, Oxford University, 1984. (2) Ottewill, R. H.; Goodwin, J. W., Ed.; Special Publication 43; Royal Society of Chemistry: 1982. (3) Auvray, L. C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim. 1986, 302, 859. (4) Auroy, P.; Auvray, L.; Le´ger, L. J. Phys.-Condensed Matter 1990, 2, SA317. (5) Auvray, L.; Auroy, P. In Neutron, X.-Ray and Light Scattering; Lindner, P., Zemb, T., Eds.; Elsevier Science Publishers B. V.: Amsterdam, 1991. (6) Auroy, P.; Auvray, L.; Leger, L. Physica A 1991, 172, 269. (7) Auroy, P.; Auvray, L.; Le´ger, L. Macromolecules 1991, 24, 2523. (8) Auvray, L.; Auroy, P.; Cruz, M. J. Physique I 1992, 2, 943. (9) Cosgrove, T.; Crowley, T. L.; Vincent, B.; Barnett, K. G.; Tadros, T. F. Faraday Symp. Chem. Soc. 1981, 16, 101. (10) Cosgrove, T.; Crowley, T. L.; Ryan, K.; Webster, J. R. P. Colloids Surf. 1990, 51, 255. (11) Cosgrove, T.; Ryan, K. Langmuir 1990, 6, 136. (12) Cosgrove, T.; Heath, T. G.; Ryan, K. Langmuir 1994, 10, 3500. (13) Forsman, W. C.; Latshaw, B. E. Polymer Engineering Science 1996, 36, 1114. (14) Ye, X.; Tong, P.; Fetters, L. J. Macromolecules 1997, 30, 4103. (15) Caucheteux, I.; Hervet, H.; Rondelez, F.; Auvray, L.; Cotton, J. P. New Trends Phys. Phys. Chem. Polym., [Proc. Int. Symp.] 1989, 63. (16) Hone, J. H. E.; Cosgrove, T.; Spahiannikova, M.; Obey, T.; Marshall, J. C.; Crowley, T. L. Langmuir 2002, 18, 855.
The contrast matching technique is less complicated, requiring shorter measuring times, fewer samples and less data manipulation. However, the layer scattering measured by this technique contains a scattering contribution from local polymer concentration fluctuations within the adsorbed layer. In a previous paper,16 it was shown that by including a parameter in the fitting process to simulate the scattering from these fluctuations, the volume fraction profile could be easily and explicitly obtained. There are two approaches to extracting the volume fraction profiles from these data, by either transform methods17 or by data fits using model volume fraction profiles. In this paper, we model SANS data at contrast match, by selecting generic and theoretical volume fraction profiles to insert into the scattering equations. Several theoretical models exist that can describe the adsorption of homopolymers onto planar surfaces. The scaling theory of de Gennes18,19 predicts a volume fraction profile with three distinct regimes. The proximal regime near the surface which has a constant polymer concentration, the semidilute, central regime which decays as φ(z) ∼ z-R ) z-4/3 (where z is the distance normal to the colloidal interface) and the distal regime, furthest from the surface which decays exponentially to the bulk concentration. Our past attempts to fit scaling profiles to neutron scattering data have proven difficult. As a pragmatic solution, exponentially decaying profiles have been used,16,20 since direct inversion of data from PEO adsorbed upon polystyrene latex have generated profiles that decay approximately exponentially.17 The Scheutjens-Fleer (SF) self-consistent mean-field theory has been shown to generate volume fraction profiles that compare well qualitatively to the profiles generated by neutron scattering; however, they have not been used to model neutron scattering data since the classical SF model does not (17) Cosgrove, T.; Vincent, B.; Crowley, T. L.; Cohen-Stuart, M. A. ACS Symposium Series 1984, 240, 147. (18) de Gennes, P. G. Macromolecules 1981, 14, 1637. (19) de Gennes, P. G. Adv. Colloid Interface Sci. 1987, 27, 189. (20) Hone, J. H. E. Ph.D. Thesis, University of Bristol, 1999.
10.1021/la035865f CCC: $27.50 © 2004 American Chemical Society Published on Web 04/22/2004
Modeling of the Fraction Profile of Polymer Layers Table 1. Details of Latex Preparation
Langmuir, Vol. 20, No. 11, 2004 4481 Table 2. Latex Dispersion Characteristics
name:
DPSL-1
DPSL-2
characteristic
DPSL-1
DPSL-2
H2O/mL H-styrene/mL D-styrene/mL styrene concentration/mol dm-3 ammonium persulfate/g ammonium persulfate concentration/mol dm-3
800 0.9 2.7 4.07 × 10-2 0.406 2.21 × 10-3
2500 1.2 5.8 2.56 × 10-2 0.895 1.57 × 10-3
particle concentration/% w/w water concentration/% w/w particle radius (PCS), rh/Å particle radius (TEM), r0/Å particle radius (SANS), r0/Å log normal width (TEM) log normal width (SANS) contrast match point/% H2O contrast match point/10-6 Å-2 ζ potential /mV
7.79 ( 0.05 5.4 ( 0.1 625 ( 20 600 ( 44 616 ( 5 0.0489
10.26 ( 0.05 5.0 ( 0.1 443 ( 10 403 ( 12
generate profiles with sufficiently high adsorbed amounts. In addition, the central part of the volume fraction profiles of the classical mean-field approximation decay with a power law exponent R ) 2, not R ) 4/3 as predicted by de Gennes. In a new constrained variation of the SF model the adsorbed amount becomes a free parameter in the generation of volume fraction profiles, enabling higher adsorbed amounts and a power law exponent close to that predicted by de Gennes are found. In this paper, the three types of volume fraction profile described above will be used in the simulation of on and off-contrast neutron scattering data. Where on-contrast data is used, a parameter will account for neutron scattering from local fluctuations in the concentration of polymer in the adsorbed layer. These new approaches will be compared and discussed in the light of theoretical predictions for adsorbed polymer layers and the limitations of small-angle neutron scattering experiments. Experimental Section Materials. Two similar substrates were used in this experiment. The first, (DPSL-1), was an approximately 75% deuterated polystyrene latex, the second, (DPSL-2), was an approximately 85% deuterated polystyrene latex. Both latices used were prepared via the surfactant-free emulsion polymerization process,21-24 the exact quantities of materials used are detailed in Table 1. The latices were dialyzed against distilled water for a minimum of 14 water changes and then against MilliQ water for at least three water changes all over a minimum period of three weeks. The latices were concentrated by rotary evaporation and centrifugation and re-dispersed in D2O (Goss Scientific Ltd. >99.9% D). The H2O content was determined by high-resolution NMR. The proton peak area of several H2O/D2O mixtures were measured and used as a calibration curve. The proton peak of the latex dispersion was then measured and compared to the calibration curve to determine the H2O content. The particle sizes of the two latices were measured in different ways. Latex DPSL-1 was measured by TEM and SANS, whereas latex DPSL-2 was measured by PCS (Malvern Autosizer 4700) and SANS. The zeta potential of both latices was measured using a Brookhaven Zetaplus electrophoresis instrument. The contrast match point, and hence scattering length density, of the latex was determined using methods described elsewhere16,25 The final dispersion characteristics are detailed in Table 2. Several samples of poly(ethylene oxide) were used in the neutron scattering experiments. PEO has a mass density of 1.13 g cm-3 and a neutron scattering length density of 0.64 × 10-6 Å-2. All samples were obtained from Polymer Laboratories Ltd.; their properties are detailed in Table 3. An adsorption isotherm of 112 kD PEO was performed twice independently upon protonated polystyrene latex. The polystyrene latex used to measure the adsorption isotherm had been prepared in the same way as the deuterated latices used in the (21) Latex LGDPS170999 was synthesised by Miss Lucie Garreau. (22) Ottewill, R. H.; Shaw, J. N. Kolloid Z. Z. Polym. 1967, 218, 34. (23) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Br. Polym. J. 1973, 5, 347. (24) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Colloid Polym. Sci. 1974, 252, 464. (25) King, S. M. In Modern Techniques for Polymer Characterisation; Pethrick, R. A., Dawkins, J. V., Eds.; Wiley: 1999.
17.0 5.2 ( 0.05 -32.0 ( 1.0
0.077 11.5 5.6 ( 0.05 -29.0 ( 1.0
Table 3. Properties of PEO PEO molecular weight, Mp/g mol-1
polydispersity/ (Mw/Mn)
batch no.
10 600 112 100 288 000 634 000
1.03 1.02 1.06 1.08
20829-3 20835-8 20840-11 20837-7
SANS experiments,22-24 and should therefore have similar surface properties. The adsorption isotherm was performed using the standard depletion method26 giving a maximum adsorbed amount of approximately 0.56 mg‚m-2 at an equilibrium concentration of 50 ppm; this is in reasonable agreement with previous measurements reported elsewhere on similar latices.20 The maximum adsorbed amount of the other molecular weight PEO samples was estimated from the work of Barnett.27 Sample Preparation. The scattering measurements were performed in three separate experiments on two different neutron scattering instruments. The samples containing 112 kD PEO were adsorbed on latex DPSL-1 whereas all other samples were adsorbed on DPSL-2. Appropriate amounts of latex were added to PEO solutions to give the required polymer coverages. The polymer in solution is below the detection limit by SANS. The stock coated latex dispersions were stored in an airtight container to minimize D2O/H2O exchange with the atmosphere. After the contrast match point of the bare latex had been determined, the contrast matched scattering samples were prepared. This was done at least 1 h before the scattering from the sample was measured. The required amount of H2O/D2O was added to the coated latex dispersion, to yield a volume fraction of φp ) 0.048. This volume fraction was chosen to maximize the signal-to-noise ratio of the scattering from the adsorbed polymer layer and yet avoid the problem of structure factors that may be encountered with more concentrated systems. Each contrast matched scattering sample therefore contained latex at a volume fraction of φp ) 0.048, a total water content of 11.5 ( 0.05% v/v and PEO concentrations of 1.95, 2.60, and 2.90 mg mL-1 for the 10.6, 288, and 634 kD PEO systems, respectively. These concentrations of PEO correspond to maximum surface coverages of 0.6, 0.8, and 0.9 mg m-2 for the 10.6, 288 and 634 kD PEO systems, respectively. Measurements. All of the experiments detailed here were performed at the ILL, Grenoble, France. However, the experiments were performed in three sessions upon two different SANS instruments (D22 and D11). These instruments have slightly different geometries and hence have different Q ranges and resolutions. Each separate experimental session used a slightly different instrument set up, resulting in different Q ranges. All samples were measured in 2 mm quartz Helma cells and were equilibrated at 298 ( 0.1 K. The sample run times varied between 1 h and 3 h, largely depending upon the sample composition and their scattering characteristics. To obtain the interference term (112 kD PEO) involves the subtraction of two off-contrast samples from one another, therefore, run times had to be unusually long so as to ensure good enough statistics in order to determine the difference accurately. The scattering from each sample was corrected for the electronic background (using B4C as a neutron absorber), detector (26) Shar, J. A. Ph.D. Thesis, University of Bristol, 1998. (27) Barnett, K. G. Ph.D. Thesis, Bristol University, 1982.
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dead time, scattering from the neutron cell (using an empty cell) and the transmission of the sample. The corrected data were then scaled to absolute intensity using the scattering from a water sample and an empty beam transmission. Data Fitting. A detailed discussion of the equations and methods used to fit the scattering from bare particles, on-contrast layer scattering and off-contrast coated particle scattering may be found in the work of Hone et al.16 Model Volume Fraction Profiles. To fit the on-contrast and interference scattering it is necessary to insert model volume fraction profiles into the scattering equations as described in recent work.16 In this work, three volume fraction profiles have been used and compared with each other. Exponential Profile. The exponential volume fraction profile is described by eq 1
( )
φ(z) ) φs exp -
z z0
(1)
where φs is the volume fraction of polymer at the interface, z is the distance normal to the interface and z0 is the decay length which controls the extension of the profile. When fitting neutron scattering data using an exponential profile, the volume fraction of polymer at the interface, φs, and the decay length of the exponential, z0 were allowed to float. Scaling Profile.18,19 The scaling profile may be separated into three distinct regimes as follows: Proximal (z ∼ a < D) where z is the distance from the interface, a is the polymer monomer length and the length, D, is defined as a , D. The volume fraction of polymer adsorbed at the interface is defined by the parameter, φs and the width of this region is defined by D. In this regime, the short-range forces between polymer segments and the interface are important with a large proportion of the polymer segments adsorbed as trains, hence the width of this region is related to the adsorption energy parameter, χs. The polymer volume fraction in the proximal region is constant. Central (D < z < D+ zsc). In this region, the adsorbed polymer volume fraction is composed of loops and tails and decays independently of the bulk concentration according to eq 2. In our model, zsc determines the extent of the central region and is a fitting parameter.
φ ≈ z-4/3
(2)
Distal (z > D+ zsc). In this region, the adsorbed polymer volume fraction is mostly comprised of tails and a few large loops, the profile decays exponentially to the bulk polymer volume fraction with a decay length, z0. When simulating neutron scattering data from adsorbed polymer layers using a scaling theory type profile, there are four parameters defining the volume fraction profile, D, φs, zsc, and z0. Where possible, when fitting the scattering from the adsorbed layers, all parameters were allowed to float, though it was often found difficult to obtain a physically reasonable profile. In such cases, either the volume fraction at the surface was fixed, or the monomer length controlling the scattering from fluctuations was held fixed since both values can be determined from fits to the data using other profiles. Further details about the fitting procedure are given in the results section. Scheutjens-Fleer Volume Fraction Profile. The SF profiles used in this paper were calculated according to a new “constrained” SF model. We consider homopolymers with segments of type A and segment ranking numbers s ) 1, ..., N, in the half-space z > 0 next to a planar surface. In each layer, we will allow two types of segments, φA(z) and φW(z), where the latter is the density of the monomeric solvent molecules. Analogous to the classical SCF approach it is assumed that there exists a dimensionless segment potential for each segment type
uA(z) ) u′(z) + χAW〈φW(z) - 1〉 + (1/6)χASδz,1 uW(z) ) u′(z) + χAW〈φA(z) - 1〉 + (1/6)χWSδz,1
(3)
where s represents the surface and where the usual normalization
of the segment potential corresponds to the condition that the polymer concentration in the bulk is set to zero. In eq 3, the first term is the Lagrange field, which is segment-type independent and coupled to the incompressibility constraint
φW(z) + φA(z) ) 1
(4)
This means that u′(z) is adjusted until eq 4 is satisfied. The second terms in eq 3 are the classical short-range nearestneighbor terms in which the well-known FH parameter between the polymer and solvent occurs. The third terms in eq 3 represent the interactions with the surface. These terms are only operational when z ) 1. In the incompressible limit, only the difference between the two parameters χAS and χWS is important. Therefore, we will set χWS ) 0 without loss of generality. We note that the Silberberg adsorption parameter is given by χs ) -(1/6) (χAS - χWS) ) -(1/6)χAS. The angular brackets in eq 3 represent a three-layer average density, which implies a non-local contribution to the segment potential
1 4 1 1 ∂2φ 〈φ(z)〉 ) φ(z - 1) + φ(z) + φ(z + 1) ≈ φ(z) + 6 6 6 6 ∂z2
(5)
where on the right-hand side the z-coordinate is taken as a continuous spatial variable. The segment potential as defined in eq 3 is used to compute the polymer and solvent density profiles. At this point, a procedure is taken that deviates from the classical one. As the polymer density in the bulk is ignored, we can no longer equilibrate the layer with bulk chains. It is important to realize that the chains that are in the adsorbed layer typically touch the surface at least with one of their segments. Therefore, it is natural to concentrate on all conformations that visit the z ) 1 coordinate at least once. We will use the super-index a to refer to this subset. The simplest way to compute the statistical weight of this fraction is to consider also the set of all conformations that never touch the surface. This subset will be referred to with the super-index f. The sum of these two sets gives the full set of conformations. The discrete version of the Edwards equation reduces in the SF-SCF formalism to a set of recurrence relations that feature chain end distribution functions G(z,s|z′,s′). Such an end-point distribution contains the combined statistical weight of all possible conformations that start with segment s′ at coordinate z′ and end with segment s at coordinate z. Typically, we will integrate (sum) over all coordinates of the starting segment z′ and consider the two end-point distribution functions which start at segment s ) 1 or at s ) N, i.e., G(z,s|1) and G(z,s|N), respectively. As anticipated above, these two end-point distributions are splitup into adsorbed and free parts, G ) Ga + Gf, where the arguments have been omitted. The recurrence relations for the complete set of conformations is given by the classical SF-SCF relations
G(z,s|1) ) 〈G(z,s - 1|1)〉e-uA(z) G(z,s|N) ) 〈G(z,s + 1|N)〉e-uA(z)
(6)
where the angular brackets again indicate a three-layer averaging similar as given above in eq 5. Each of these recurrence relations needs a starting condition. Realizing that a walk of just one segment has the statistical weight of a free detached segment we can write G(z,1|1) ) G(z,N|N) ) exp(-uA(z)). The f-contribution is given by the combined statistical weight of those conformations that do not visit the z ) 1 coordinate. This is computed by dismissing all the contributions that originate from the first layer. The starting conditions will be set to Gf(z,1|1) ) Gf(z,N|N) ) (1 - δz,1) exp(- uA(z)) and the propagators have almost the same structure as in eq 6 except in the surface layer
Gf(z,s|1) ) 〈Gf(z,s - 1|1)〉e-uA(z)(1 - δz,1) Gf(z,s|N) ) 〈Gf(z,s + 1|N)〉e-uA(z)(1 - δz,1)
(7)
The difference between the overall- and free end-point distributions gives the statistical weight of all conformations that touch
Modeling of the Fraction Profile of Polymer Layers
Langmuir, Vol. 20, No. 11, 2004 4483
Figure 1. Log-log plot of calculated SF profiles at full surface coverage for 10.6 kD (full line), 112 kD (dotted line), 288 kD (dashed line) and 634 kD (dash-dot-dot-dash) PEO. The gray line represents a z-4/3 decay.
Figure 2. Interference scattering from 112 kD Mw PEO adsorbed at full surface coverage upon polystyrene latex (open circles) fitted with an exponential volume fraction profile (solid line), a scaling theory profile (dashed line) and a scaled Scheutjens-Fleer profile (dash-dot-dot-dash).
the surface at least once, i.e., G ) Ga + Gf. The density profile due to this class of conformations follows from the composition law N
∑e
φ(z) ) C
ua
Ga(z,s|1)Ga(z,s|N)
(8)
s)1
The contribution due to free chains is thus completely dismissed. In eq 8 the two complementary distribution functions overlap with segment s at coordinate z. Close inspection reveals that in both distribution functions the segment potential is included for segment s. It is necessary to correct for this by including the factor exp(uA(z)) in eq 8. The last point is to discuss the normalization constant C. The value of this constant is a free parameter in the present model and it controls the total amount of adsorbed polymer θ
θ)
∑φ(z)
(9)
z
Note that the exact value of C depends on how the segment potentials are normalized. This means that the normalization of the segment potentials becomes irrelevant for the final result. For completeness, we mention that the distribution of the solvent molecules follows directly from the fact that the solvent molecules are monomers: φW(z) ) exp(-uW(z)). The normalization is unity in this case, because the bulk concentration of solvent is unity. The parameters used to generate the profiles are taken from the literature.11 The ratio of polymer monomers to statistical polymer segments, P/r, was taken as approximately 4 (i.e., 4 PEO monomers per segment). The Flory-Huggins segmentsolvent interaction parameter, χ, was taken to be the same as the experimentally determined value for PEO in water (0.45).28 The net surface-segment interaction parameter, χs, for PEO on polystyrene latex made with persulfate initiator was taken as 0.6.11 The profiles generated by the constrained SF model are shown in Figure 1. To simulate the SANS data using the calculated profiles, the z and φ(z) axis of the profiles were re-scaled. The z-axis was re-scaled by assigning a specific length (in Å) per lattice unit; the φ(z) axis was multiplied by a scaling parameter obtained through fitting. The calculated profile for 112 kD PEO at full surface coverage was re-scaled to simulate the interference scattering from 112 kD PEO at full surface coverage. These scaling parameters were used to re-scale the other calculated volume fraction profiles at different molecular weights. (28) Bandrup, J.; Immergut, E. H. Polymer Handbook; WileyInterscience: London, 1975.
Figure 3. Q3I(Q) against Q plot of interference scattering from 112 kD Mw PEO adsorbed at full surface coverage upon polystyrene latex (open circles) fitted with an exponential volume fraction profile (solid line), a scaling theory profile (dashed line) and a scaled Scheutjens-Fleer profile (dash-dotdot-dash).
Results Approach to Determining Volume Fraction Profiles. Volume Fraction Profiles of 112 kD PEO. The 112 kD intermediate molecular weight of PEO is dealt with first since the scattering from this system has been measured in two ways. It has been shown that it is possible to obtain unambiguously volume fraction profiles for adsorbed polymer layers from the interference scattering and from the on-contrast layer data.16 When simulating the on-contrast layer scattering, the model must account for the scattering arising from concentration fluctuations in the adsorbed polymer layer, while the interference scattering contains no contribution from concentration fluctuations. Figure 2 shows the interference scattering from 112 kD PEO adsorbed upon latex DPSL-1. The scattering data have been simulated with each of the three model profiles detailed earlier, with smearing for polydispersity and resolution effects. Figure 3 shows the same scattering and fits plotted as Q3I(Q) against Q, the form of this plot highlights differences between the data and the fits. When fitting the scattering from systems adsorbed upon latex DPSL-1, the log-normal width of the particle size distri-
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Table 4. Parameters Used to Fit the Interference Scattering from 112 kD PEO Adsorbed upon Polystyrene Latexa parameter proximal layer thickness, D/Å width of scaling region, zsc/Å decay length, z0/Å resolution width, q/10-4 Å-1 log-normal width, σs particle size, rp/Å incoherent background, Iinc/ 10-3 cm-1 volume fraction of polymer at interface, φs φ scaler z scaler/Å χ2 a
exponential profile value
scaling profile value
24.8 ( 0.8 (6.3) (0.0489) (623) 4.6 ( 0.2 0.131 ( 0.08
9.1 ( 0.4 75.1 ( 251.6 43.8 ( 131.9 (6.3) (0.0489) (623) 3.2 ( 0.1 (0.131)
64
34
Scheutjens-Fleer profile value
(6.3) (0.0489) (623) 3.8 ( 0.1 0.24 ( 0.01 8.46 ( 0.26 82
Parameters enclosed in brackets were fixed. Table 5. Physical Parameters Derived from Fits to the Interference Scattering from 112 kD PEO at Full Surface Coverage Adsorbed upon Polystyrene Latex parameters
exponential profile
scaling theory profile
Scheutjens-Fleer theory profile
adsorbed amount, Γ/mg m-2 RMS thickness, δRMS/Å volume fraction at the interface, φs
0.37 ( 0.03 35.9 ( 1.1 0.131 ( 0.008
0.38 ( 0.06 39.2 ( 57.0 (0.131)
0.40 ( 0.02 52.7 ( 1.6 0.138 ( 0.006
Figure 4. Semilog plot of the volume fraction profiles derived from fits to the interference scattering from 112 kD PEO at full surface coverage using an exponential profile (solid line) a scaling profile (dashed line) and a scaled Scheutjens-Fleer profile (dash-dot-dot-dash). The radius of gyration (Rg) for 112 kD PEO as calculated from PCS data17 is also shown.
bution was fixed at 0.0489 as determined from TEM size distribution measurements and is in good agreement with fits to the bare particle scattering. The instrument resolution for the interference scattering was fixed at 6.3 × 10-4 Å-1 in order to reduce the number of free parameters. This value was chosen as an average value obtained from fits where the instrument resolution was allowed to float. The parameters arising from fits to the interference scattering are detailed in Table 4. The volume fraction profiles derived from fits to the interference scattering are shown in Figure 4 and the physical parameters of the profiles are presented in Table 5. Figure 5 illustrates the on-contrast scattering from 112K PEO shown as Q2I against Q adsorbed at full surface coverage upon polystyrene latex fitted using an exponential profile, a scaling theory profile and a ScheutjensFleer theory profile as detailed in previously. The fits to the on-contrast scattering include a parameter, the monomer length, to account for the scattering from polymer concentration fluctuations in the adsorbed polymer layer. The parameters arising from fits to the oncontrast scattering are detailed in Table 6. The volume fraction profiles derived from fits to the on-contrast scattering are shown in Figure 6 (black lines), plotted alongside the volume fraction profiles obtained from fits
Figure 5. Q2I(Q) against Q plot of the scattering from 112 kD Mw PEO adsorbed at full surface coverage upon polystyrene latex (open circles) fitted with an exponential volume fraction profile (solid line), a scaling theory profile (dashed line) and a scaled Scheutjens-Fleer profile (dash-dot-dot-dash).
to the interference scattering (gray lines). The physical parameters of the profiles are presented in Table 7. It should be noted that when fitting the on-contrast layer scattering using the scaled Scheutjens-Fleer profile, the same scaling parameters as obtained from the fit to the interference scattering were used, thus only the incoherent background and the monomer length were allowed to float. When using the scaling profile to fit the scattering data, large errors were obtained in predicting the end of the central (self-similar) region and the start of the distal (exponential) region. This arises from the low sensitivity of neutrons in the tail region of the profile,17,29 due to the low volume fraction of polymer (φp e 0.01). As a result, there is quite a large uncertainty in both the width of the central region (where φ ∼ z-4/3) and on the extent of the tails (exponential region). To address this issue, the approach we adopted was to assign the value of the central region to that obtained from the SF model. It was then allowed to float. The scaling model being very insensitive to the tail region, the extent of the exponential decay was also fixed to a reasonable (small) value. Once a value for the width of the central region was obtained after (29) Vaynberg, K. A.; Wagner, N. J.; Sharma, R.; Martic, P. J. Colloid Interface Sci. 1998, 205, 131.
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Langmuir, Vol. 20, No. 11, 2004 4485
Table 6. Parameters Used to Fit the On-Contrast Layer Scattering from 112 kD PEO Adsorbed upon Polystyrene Latexa parameter proximal layer thickness, D/Å width of scaling region, zsc/Å decay length, z0/Å resolution width, q/10-4 Å-1 log-normal width, σs particle size, rp/Å incoherent background, Iinc/10-3 cm-1 monomer length, a/Å volume fraction of polymer at interface, φs φ scaler z scaler/Å χ2 a
exponential value
scaling theory value
25.5 ( 1.2 (6.8) (0.0489) (638) 2.2 ( 0.1 2.15 ( 0.05 0.136 ( 0.07
9.3 ( 1.0 (120) (15) (6.8) (0.0489) (638) 2.1 ( 0.1 (2.22) 0.140 ( 0.01
0.95
1.4
Scheutjens-Fleer value
(6.8) (0.0489) (638) 2.0 ( 0.1 2.27 ( 0.02 (0.24) (8.46) 1.34
Parameters enclosed in brackets were fixed.
Table 7. Physical Parameters Derived from Fits to the On-contrast Scattering from 112K PEO at Full Surface Coverage Adsorbed upon Polystyrene Latexa parameters
exponential profile
scaling theory profile
Scheutjens-Fleer theory profile
adsorbed amount, Γ/mg m-2 RMS thickness, δRMS/Å volume fraction at the interface, φs
0.39 ( 0.02 36.8 ( 0.4 0.14 ( 0.07
0.41 ( 0.01 (42.3) 0.14 ( 0.01
(0.40) (46.7) (0.129)
a
Parameters enclosed in brackets were fixed.
Figure 6. Semilog Comparison of volume fraction profiles obtained from fits to the on-contrast layer scattering (black lines) and interference scattering (grey lines) for 112 kD PEO at full surface coverage. The exponential profile is represented with the solid line, the scaling profile with the dashed line and the scaled SF profile with the dash-dot-dot-dash. The calculated radius of gyration (Rg) for 112 kD PEO17 is also shown. For the SF profile, both the on- and off-contrast profiles are identical.
convergence of the fit, the length of the decay region, z0, was allowed to adjust (in practice, it did not vary much from the input value). As in our previous paper,16 the agreement between the sets of profiles obtained from these two independent SANS methods is remarkable. It should be noted that the χ2 values quoted for the interference scattering fits appear large and very different between fits. This is because the data has few points and relatively large oscillations compared to the intensity of the scattering. For this molecular weight, the exponential volume fraction profile appears to provide a very good fit to the SANS data. This profile appears to be a rough “average” of the more intricate scaling and Scheutjens-Fleer fits. The parameters used to re-scale the lattice unit size of the SF profile are close to the expected value for four PEO monomer units. The adsorbed amount obtained for the three profiles is the same within experimental error but slightly lower
than that obtained on a similar latex by the depletion method. The δRMS obtained for the exponential profile however is smaller than that obtained for the SF and scaling theory profiles. This is because the SF profile accounts for the polymer tails, whereas the profile obtained directly from fitting the SANS data does not. In the case of the scaling profile, the width of the central region has been adjusted to the one obtained from the SF model. For this reason, the δRMS obtained in this case is comparable to the one predicted by the SF model. Comparison of the volume fraction profiles with the radius of gyration (Figure 6) shows that the exponential profile predicts a maximum layer extension just short of the radius of gyration (Rg) while the scaling and SF profiles predict an extension comparable to the Rg. Volume Fraction Profiles of 10.6 kD PEO. When fitting the 10.6 kD PEO, the parameters obtained from the 112 kD data were used for the Scheutjens-Fleer profile. The same parameters were allowed to float for the 10.6 kD as were with the 112 kD data. The physical parameters derived from the volume fraction profiles are presented in Table 8. The three volume fraction profiles obtained are clearly very similar, once again all three have similar adsorbed amounts, however, the δRMS for the SF profile is smaller than for the exponential and scaling profiles. This may be because the tail region of this low molecular weight PEO is small and therefore insignificant in the calculation of δRMS. In addition, the central, scaling region of the SF and scaling volume fraction profiles is very small or nonexistent, this is better illustrated in the log-log plot of Figure 7. Volume Fraction Profiles of 288 kD PEO. The data for this sample has lower resolution, poorer statistics at high Q and a much smaller Q range. This made fitting the data problematic; consequently, it was necessary to fix the monomer length in the scaling theory simulation in order to obtain volume fraction profiles that were physically reasonable. As before, the width of the central region was first given the value obtained from the SF model and then allowed to float. The characteristics of the volume fraction profiles are detailed in Table 9. As with the volume fraction profiles derived from the 112 kD PEO scattering data, the exponential profile has
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Table 8. Physical Parameters Derived from Fits to the On-Contrast Scattering from 10.6 kD PEO at Full Surface Coverage Adsorbed upon Polystyrene Latex parameters
exponential profile
scaling theory profile
Scheutjens-Fleer theory profile
adsorbed amount, Γ/mg m-2 RMS thickness, δRMS/Å volume fraction at the interface, φs
0.25 ( 0.04 21.4 ( 0.3 0.149 ( 0.012
0.24 ( 0.48 21.2 ( 11.6 0.120 ( 0.070
0.27 15.8 0.117
Table 9. Physical Parameters Derived from Fits to the On-contrast Scattering from 288 KD PEO at Full Surface Coverage Adsorbed upon Polystyrene Latex parameters m-2
adsorbed amount, Γ/mg RMS thickness, δRMS/Å volume fraction at the interface, φs
exponential profile
scaling theory profile
Scheutjens-Fleer theory profile
0.39 ( 0.15 35.1 ( 4.5 0.143 ( 0.025
0.50 ( 0.46 100 ( 40 0.131 ( 0.07
0.43 69.2 0.129
Volume Fraction Profiles of 634 kD PEO. To simulate the 634 kD data using the SF volume fraction profile, it was necessary to allow the scaler for the volume fraction to float rather than fixing it at the value obtained from the 112 kD PEO interference scattering data. However, there is little difference (0.26 instead of 0.24) between the value used for this molecular weight and the others. Once again, all other parameters were obtained from the 112 kD data. The physical characteristics of the volume fraction profiles are detailed in Table 10. As with the 288 kD PEO data, the adsorbed amount and δRMS for the SF and scaling volume fraction profiles are very similar. As before, this may be explained by the inclusion of the tail region of the adsorbed polymer layer in the SF profile and the fact that this profile has been used to determine the scaling profile. This can be illustrated by the comparison of the volume fraction profiles with the radius of gyration and suggests SANS sensitivity at the extremities of the polymer layer, beyond the region containing both loops and tails,17,29 is lower. Figure 7 provides a direct comparison of all the profile types at all molecular weights. Discussion Figure 8 shows the calculated δRMS for each type of volume fraction profile at each molecular weight. The data have been fitted with a power law as given in eq 10.
δRMS ∼ MwR
Figure 7. Semilog plots of the (a) exponential, (b) scaling and (c) Scheutjens-Fleer volume fraction profiles determined from the SANS of 10.6 kD PEO (solid line), 112 kD PEO (dotted line), 288 kD PEO (short dash line) and 634 kD PEO (dashdot-dot-dash line) adsorbed on PSL. Also plotted is φ(z) ∼ z-4/3 line (thin solid line).
a δRMS much smaller than the scaling and SF models. The adsorbed amount also is lower, this is due to the tail region which is not properly accounted for by the exponential profile, as explained above. This appears clearly in the semilog plot of the volume fraction profiles and the comparison with the radius of gyration Rg. It should be noted that the fits using the scaling profile are heavily dependent upon the quality of the data. Here we had to use the predictions of the SF model in order to obtain good fits, but with data of higher resolution over a larger Q range, we would probably have been able to obtain from the fits profiles similar to the SF model (with a greater extension of the tails into the bulk solution).
(10)
where R is the fitted exponent. The calculation of δRMS for the SF profiles in the absence of the tail region is also plotted. This was estimated by excluding any of the volume fraction profile below φ(z) ) 0.003 (approximately where the scaling regions of the higher molecular weight polymers end).The exponents obtained from fits to the δRMS against molecular weight data are listed in Table 11. The dependence of δRMS upon Mw measured by the three profiles shows the importance of including the tail region in the calculation. The small exponent reported for the exponential profile is in disagreement with previous SANS measurements17 of δRMS where it was seen to obey δRMS ∼ Mw0.4. This is possibly because the volume fraction profile derived from the SANS measurements performed by Cosgrove et al.17 were made via a direct Hilbert transformation of the scattering data extrapolated at high and low Q. The volume fraction profiles were normalized by adsorbed amounts measured by a different, independent method (which includes the tail region) not from adsorbed amounts determined directly from SANS. Also, the extrapolation back to high and low Q, may help to create profiles with larger tail regions than were actually detected by SANS. The δRMS dependence of the calculated SF
Modeling of the Fraction Profile of Polymer Layers
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Table 10. Physical Parameters Derived from Fits to the On-contrast Scattering from 634 KD PEO at Full Surface Coverage Adsorbed upon Polystyrene Latex parameters
exponential profile
scaling theory profile
Scheutjens-Fleer theory profile
adsorbed amount, Γ/mg m-2 RMS thickness, δRMS/Å volume fraction at the interface, φs
0.43 ( 0.02 39.0 ( 1.0 0.134 ( 0.004
0.51 ( 0.13 130 ( 18.9 0.140 ( 0.01
0.49 104.3 0.142
Figure 8. Log-log plot of δRMS of PEO as a function of molecular weight for exponential profiles (open squares), scaling profiles (open circles) and scaled Scheutjens-Fleer profiles (closed triangles). The solid, dashed, dash-dot-dot-dash and dotted lines are power law fits to the exponential, scaling, ScheutjensFleer and truncated Scheutjens-Fleer data, respectively. Also plotted is the δRMS for the Scheutjens-Fleer profiles where the tail region has been omitted from the calculation of δRMS by excluding data below a volume fraction of 0.003 (open diamonds). Table 11. Power Law Exponent for Fits to the δRMS against Molecular Weight for the three types of profiles volume fraction profile
δRMS ∼ Mwx/x
exponential scaling Scheutjens-Fleer truncated Scheutjens-Fleer
0.15 0.45 0.46 0.27
weight of the PEO used. This result is similar to that obtained by Barnett27 where an exponent of only 0.06 was found for the molecular weight dependence of adsorption of PEO upon polystyrene latex. These observations are consistent with the relatively good solvency condition for PEO in water. Conclusions
profiles presented here is slightly larger than δRMS ∼ Mw0.4 as estimated by de Gennes.19 Figure 8 shows that when the tail region (φ < 0.003) of the SF profile is removed, the calculated δRMS closely resembles that of the exponential profile obtained through fitting on-contrast neutron scattering data. Previous dynamic light scattering data30 also showed the importance of the tail region in determining the hydrodynamic layer thickness, which is substantially greater than δRMS. Figure 9 shows the adsorbed amount of PEO as a function of molecular weight as determined by the three types of volume fraction profile. The trends in adsorbed amount determined by the three profiles are similar. However, slight differences occur at high molecular weights, as observed with δRMS, as the exponential profiles predict adsorbed amounts slightly lower than those determined from the SF and scaling profiles. However, the difference between the profiles is much smaller than in the case of δRMS. From the slope of the straight line fitted to all the data sets combined, the following relationship is obtained
Γ ∝ Mw0.15
Figure 9. Amount of PEO adsorbed upon polystyrene latex as a function of molecular weight for three profiles, exponential profile (open circles), scaling profile (open squares) and Scheutjens-Fleer profiles (closed triangles).
(11)
This indicates that the amount of polymer adsorbed at the interface has a weak dependence upon the molecular (30) Cohen Stuart, M. A.; Waajen, F. H. W. H.; Cosgrove, T.; Vincent, B.; Crowley, T. L. Macromolecules 1984, 17, 1825.
Three different model volume fraction profiles have been used to simulate the interference and on-contrast layer scattering from PEO of different molecular weights and surface coverage adsorbed upon polystyrene latex. The three volume fraction profiles used were an exponential decay, a profile based on scaling theory and profiles calculated using a new constrained model of the Scheutjens-Fleer self-consistent mean field theory. When simulating the interference scattering and on-contrast layer scattering of 112 kD PEO, very similar profiles were generated by the two SANS methods. As in a previous paper,16 this demonstrates that fitting of on-contrast layer scattering with a parameter to model the scattering from local concentration fluctuations in the polymer layer explicitly generates the correct volume fraction profiles. When simulating the scattering from systems with low molecular weight PEO, all three model profiles are very similar, with similar adsorbed amounts and layer thicknesses. As the molecular weight of PEO in the systems increases, the scaling and SF profiles remain very similar in the proximal and central (self-similar) regions. The exponential profile begins to “average” the more intricate details within the scaling and SF profiles in the proximal and central regions. The main result of these experiments is that the new constrained Scheutjens-Fleer mean-field calculations can predict the structure and hence the scattering of adsorbed PEO layers with great accuracy. The three profiles differ in the distal region of the profiles, where the tail fraction of the adsorbed layer dominates. The calculated SF profile takes greater account of the tails in the volume fraction profiles. Since the scaling and exponential volume fraction profiles are directly fitted
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to the scattering data, the insensitivity of neutrons to the tail region of the profile means that these models may be truncated in the distal region and may not account for the entire tail region. In the case of the scaling profile, this was corrected for by using the value obtained from the SF predictions for the width of the central region. Therefore the values obtained for δRMS and Γ are quite similar for both models. At high molecular weights (>100 kD PEO), the difference between the various models increases. Consequently, measurements of δRMS become increasingly different while Γ remains only slightly different. Where the tail region is excluded from the calculated SF profiles, measurements
Marshall et al.
of δRMS are virtually identical to the fitted exponential profiles. This indicates the limitations of obtaining volume fraction profiles by directly fitting the small angle neutron scattering from high molecular weight polymer layers. Acknowledgment. We would like to thank Dr. Isabelle Grillo for her help with the ILL experiments. J.M. would like to acknowledge Kodak European Research and Development and the EPSRC for funding. CD would like to acknowledge the EPSRC Impact Faraday Partnership and ACORN for the provision of a PDRA fellowship. LA035865F
