Evidence of a Sticky Boundary Layer in Nanochannels: A Neutron

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Evidence of a Sticky Boundary Layer in Nanochannels: A Neutron Spin Echo Study of n-Hexatriacontane and Poly(ethylene oxide) Confined in Porous Silicon Andr e Kusmin,*,† Simon Gruener,z,‡ Anke Henschel,‡ Olaf Holderer,§ €rgen Allgaier,† Dieter Richter,† and Patrick Huber*,‡ Ju †

Institut f€ ur Festk€ orperforschung, Forschungszentrum J€ ulich, 52425 J€ ulich, Germany, ‡Experimental Physics, § ulich Centre for Neutron Science, c/o TU M€ unchen, Saarland University, 66041 Saarbr€ ucken, Germany, and J€ Lichtenberg Str. 1, 85747 Garching, Germany

ABSTRACT Neutron spin echo spectra of the melts of n-hexatriacontane (C36) and poly(ethylene oxide) (PEO), 2 kg/mol, confined in circular channels with a mean diameter of 10 and 12 nm, respectively, in porous silicon, were recorded at Q values between 0.05 and 0.15 Å-1. The spectra were successfully analyzed in terms of a two-state model where wall-adsorbed molecules are immobile and free molecules have a bulk-like dynamics. For C36, we find an adsorbed bilayer at 364 K and a monolayer at 435 K (in both cases, the long axis of the molecules is oriented parallel to the surface) and no adsorbed layer at 512 K. For PEO, we find an adsorbed monolayer at 413 K. The results support the existence of a sticky boundary layer inferred from capillary filling experiments. SECTION Macromolecules, Soft Matter

C

In porous silicon, the surface of the pores is covered with SiO2. Alkane molecules adsorb on this surface via van der Waals interactions and possibly via CH 3 3 3 O hydrogen bonds.15-17 PEO adsorbs even more strongly via hydrogen bonds from ether oxygens on the polymer to silanol groups.18 Alkane molecules adsorb with their long axis oriented parallel to the surface; such a monolayer is ∼5 Å thick.19,20 The upper bound of the thickness of an adsorbed PEO monolayer is the root-mean-square end-to-end distance (RF) of a polymer chain in a melt, which is lN0.5, where N is the number of freely jointed segments and l is the segment length. For PEO 2k, RF is 4 nm (l2 = 33.75 Å2).21 Because the pore diameters for PEO- and alkane-filled wafers are 12 and 10 nm, respectively, the fraction of adsorbed molecules is significant. Therefore, a distinction between the dynamics of free and adsorbed molecules has to be made. We assume that whereas free molecules have a bulk-like dynamics and their centers of mass (CMs) diffuse within a narrow cylinder of a radius RD in the center of the pore, the CMs of adsorbed molecules are immobile. The lifetime of the adsorbed state (τ0) depends on the strength of surface interactions and is expected to be much longer for PEO than for n-C36H74. Therefore, whereas for PEO we assume that τ0 is longer than the observation time (τobsd) (which is 20 ns in our experiment), for n-C36H74, one should expect that τ0 is at least several times shorter than τobsd.

apillary filling is essential for the implementation of many current and potential applications of nanoporous materials;1,2 accordingly, it has been studied both by experiment3-6 and theory7 for more than a decade. Specifically, when the size of the building blocks of the invading liquids is comparable to the pore radius, an existence of a layer of adsorbed molecules was inferred.5 Microscopic investigations of the structure and dynamics of this adsorbed layer would help to resolve some controversies regarding the flow in confined systems, particularly, the validity of the no-slip boundary condition.8,9 Neutron spin echo (NSE) and quasielastic neutron scattering (QENS) are excellent methods for such investigations, as demonstrated by the studies of the dynamics of alkanes in zeolites10 and in mesoporous silica11 and the dynamics of polymer melts in alumina membranes12 and porous silicon.13 Here we present an NSE experiment on melts of n-C36H74 and poly(ethylene oxide) (PEO) with a molecular weight of 2 kg/mol (2k) confined in noninterconnected circular channels aligned strictly perpendicularly to the surface of the porous silicon wafers. Both molecules are several times smaller than the mean channel diameter; hence, filling of the porous matrix is easy, and the decay of the dynamic structure factor is sufficiently pronounced. Because of the well-defined porous space, by orienting the wafers relative to the incident beam, one can selectively probe molecular motions along the pore and in the radial direction, and, for long observation times reachable with NSE, the spectra can be interpreted without complications brought about by the disordered structure of the porous host (e.g., by the tortuosity of glasses14).

r 2010 American Chemical Society

Received Date: September 1, 2010 Accepted Date: September 29, 2010 Published on Web Date: October 12, 2010

3116

DOI: 10.1021/jz1012406 |J. Phys. Chem. Lett. 2010, 1, 3116–3121

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eq 4 with the model of the long-range translational diffusion2 23 to account for the CM dynamics, namely, Scm(Q,t) = e-DQ t, as well as with eq 3 and the condition B(Q) = 0 yields

Consequently, whereas for PEO the spectra are the sums of contributions from the dynamics of free and adsorbed molecules, the spectra of n-C36H74 reflect the average (over two states) dynamics of an alkane molecule. Specifically, the measured diffusion coefficient (D) will be smaller than the diffusion coefficient of free alkane molecules (Dbulk) according to ð1Þ

We fitted eq 5 to the bulk n-C36H74 spectra with D(T), τint(T), A(Q), and B(Q,T) being free fit parameters. The uncertainties in fitted values of A(Q) were larger than the Q-dependence of the values themselves; therefore, all fits were redone with A(Q) being a Q- independent fit parameter; that is, A(Q) = A. On the basis of resulting τint values, we chose τmin values such that Sint(Q,t)/S(Q) ≈ A for t > τmin and fitted eq 5 again, but with 1/τint set to ¥ and only at t > τmin. This restriction eliminates an uncertainty as to whether the t-dependence of Sint(Q,t)/S(Q) is adequately described by eq 4. Fit results are given in Table 1, experimental and fitted curves are shown in Figure 1A-C. Let us justify the use of eq 4 and the restriction of the time range to t > τmin. First, an intrinsically similar approach applied to the analysis of QENS spectra of n-C16H34 (with an observation time of ∼1 ns) resulted in a correct diffusion coefficient.24 Second, for the self-diffusion of n-C36H74, the activation energy (Ea) calculated from an empiric equation is 22.7 kJ/mol,25 and D = 20 [10-7 cm2/s] was measured by the pulse-field gradient NMR26 at 355 K. Using the Arrhenius law, from Table 1, we obtained Dbulk(355 K) = 14.9 [10-7 cm2/s] and Ea = 20.5 ( 0.4 kJ/mol. The agreement is satisfactory if one recalls that a slower diffusion is expected for deuterated n-C36H74 molecules. To the spectra of confined n-C36H74, we fitted eq 3, where cm Sint free(Q,t)/Sfree(Q) was given by eq 4, S (Q^,t) was given by the model of diffusion inside a circle of a radius RD,13,27 and Scm(Q ,t) was given by the model of the long-range transla2 tional diffusion, Scm(Q ,t)) = e-D Q t. We assumed that D = D^ = D and that the dynamics of free n-C36H74 molecules is bulk-like, and hence τmin and A values obtained for the bulk n-C36H74 were used. Examples of the simultaneous fit to all spectra recorded at 0 orientation are shown in Figure 1D-F; fitted values are given in Table 1. The difference between the mean pore radius Rh (Rh = 50 Å) and RD is the thickness of an adsorbed layer. It is 6 Å and thus suggests the presence of a monolayer of adsorbed molecules. The uncertainty in the fitted RD value makes this suggestion questionable. As can be seen from eq 1, a clear evidence of the existence of an adsorbed layer is represented by D/Dbulk values that are smaller than unity. Moreover, because the fraction of free molecules at any given moment is f = τ1/(τ1 þ τ0), f is just D/Dbulk. The temperature dependence of f values agrees with weakening of alkane-surface interactions with increasing temperature and with increasing mobility of free molecules. (The former leads to a decrease in τ0; the latter leads to a decrease in τ1.) Note that f ≈ 1 at 512 K indicates that there is no adsorbed layer at all. The fraction of free molecules also depends on the structure of adsorbed layer. From Rh = 50 Å and purely geometrical considerations, for a 5 Å thick monolayer, f is 0.81; for a bilayer, f is 0.64. These estimates are in a remarkable agreement

where l2 is the mean square displacement of the molecule and τ1 is the lifetime of free state. To construct the equation to be fitted to the NSE spectra, we start with the assumption that an exchange between the adsorbed and free states can be neglected (PEO case, τ0 > τobsd) and write SðQ, tÞ fSfree ðQ, tÞ þ ð1 - f ÞSads ðQ, tÞ þ ISi ðQÞ ¼ SðQÞ fSfree ðQÞ þ ð1 - f ÞSads ðQÞ þ ISi ðQÞ

ð2Þ

)

SðQ, tÞ ¼ SðQÞ

int Sfree ðQ, tÞ þ BðQÞ Sfree ðQÞ 1 þ BðQÞ

)

)

)

)

where f is the fraction of free molecules and subscripts free and ads refer to the molecules in the free and the adsorbed states, respectively. The coherent elastic scattering from Si wafers, ISi(Q), is not negligible because the scattering length densities (SLDs) of PEO and n-C36H74 were not matched to that of the Si matrix. We assume that the CM motions directed along and perpendicularly to the pore may be decoupled from each other and from the internal dynamics and that the contribution of the internal molecular dynamics is orientation independent. Consequently, for every molecule, S(Q,t) = Scm(Q ,t)Scm(Q^,t)Sint(Q,t), where Q and Q^ are the Q components directed along and perpendicularly to the pore axis, respectively. The superscripts cm and int refer to the contributions due to the CM and internal molecular dynamics, respectively. Because the confinement is not strong, we assume that the conformation of adsorbed molecules is the same as that in bulk, that is, Sfree(Q) = Sads(Q). As discussed elsewhere,13 at low Q values, the contribution of the internal dynamics of adsorbed molecules can be neglected for the lowmolecular-weight polymer and even more so for n-C36H74. Therefore, because the CM of adsorbed molecules is immobile, it follows that Sads(Q,t)/Sads(Q) = 1. With all of these assumptions, eq 2 can be rewritten as Scm ðQ , tÞScm ðQ^ , tÞ

ð3Þ

where B(Q) = (1 - f þ ISi(Q)/Sads(Q))/f. For n-C36H74, that is, when the exchange between adsorbed and free states is fast, there is, effectively, only one state. Hence, eq 3 is still valid, but f = 1 and B(Q) = ISi(Q)/Sads(Q) should be used. Let us start with the analysis of bulk n-C36H74 spectra. To account for the internal molecular dynamics, we use a phenomenological approach successfully previously applied22 Sint ðQ, tÞ ¼ AðQÞ þ ½1 - AðQÞe - t=τint SðQÞ

ð4Þ

where A(Q) is an analog of the elastic incoherent structure factor (EISF) and τint is an effective internal relaxation time that accounts for all internal dynamics. The combination of


ð5Þ

3117

)

τ1 =ðτ1 þ τ0 Þ

)

bulk

)

D ¼ l =ð6ðτ1 þ τ0 ÞÞ ¼ D 2

SðQ, tÞ 2 ¼ ðAðQÞ þ ð1 - AðQÞÞe - t=τint Þe - DQ t SðQÞ


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Table 1. Results of the Fits of the Bulk (eq 5) and Confined (See the Text) Alkane Modelsa T [K]

τint [ns]

τmin [ns]

364

0.33 ( 0.2

2

435

0.12 ( 0.06

1

512

0b

0.75

A

Dbulk [10-7 cm2/s] 17.6 ( 0.25

11.7 ( 0.8

0.66

0.893 ( 0.007

53.7 ( 0.89

42.9 ( 2.3

0.8

115.8 ( 7.4

0.98

117.9 ( 2.2

D [10-7 cm2/s]

D/Dbulk

RD [Å] 43.8 ( 8.4

a Fitted values are given with fit uncertainties. τint(T) is an internal relaxation time, Dbulk(T) and D(T) are the translational diffusion coefficients of n-C36H74 in the bulk and in confinement, respectively. A is an EISF-like quantity, RD is the radius of a cylinder the CM diffusion is confined to. τint(T) was obtained from the fit of eq 5 in the whole time range. All other parameters were fitted in the time range t g τmin(T)with 1/τint set to ¥. b Value could not be determined by fitting.

Figure 1. Examples of fitting of (A-C) eq 5 to the spectra of bulk n-C36H74 and (D-F) the confined alkane model to the spectra of confined nC36H74, 0 orientation. In the subfigure F, the fit to the spectrum recorded at 48 orientation is shown as well.

with observed f values (Table 1). (Qualitatively, the agreement is not influenced by the pore radius distribution, as seen from f values calculated from R = Rh ( hwhm (half width at half-maximum) of the distribution.) It is known that at temperatures below and slightly above the melting temperature there is a bilayer where the long axes of the alkane


molecules are oriented parallel to the flat surface.20,28,29 Our results show the existence of such a bilayer at 15 K above the bulk melting temperature (∼349 K30), in agreement with observations for n-alkane adsorption on planar silicon. One may speculate that despite the roughness of the channels the highly curved, cylindrical geometry may additionally induce

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whereas nonadsorbed molecules diffuse inside a cylinder in the center of the pore. For the alkane, from the comparison of diffusion coefficients for the bulk and confined molecules, the structure of the adsorbed layer was found to change with increasing temperature from a bilayer where molecular long axes are oriented parallel to the surface at 364 K to a monolayer at 435 K, followed by the complete disappearance of adsorption layer at 512 K. The CM diffusion of confined n-C36H74 was found to be isotropic. For PEO, an adsorbed monolayer was found. Therefore, the results support the existence of a layer of adsorbed molecules, which was previously inferred from macroscopic experiments.5 In general, NSE experiments allow for the determination of the thickness of the adsorbed layer in the porous media with both strong and weak liquid-surface interactions and complement the macroscopic experiments on the liquid flow in confinement.

Figure 2. Fit of the confined polymer model to the spectra of confined PEO 2k.

a parallel alignment of the molecules' long axes along the axial channel direction. This can, however, not be inferred from the data sets collected in our experiments. We also fitted the confined alkane model only to the spectra recorded at 512 K and Q = 0.08 Å-1, at both 0 and 48 orientation. As can be seen from the 48 orientation curve in Figure 1 F, the fitting quality is satisfactory. When we fitted the same model but with D and D^ being two independent free fitting parameters, the fit quality improved little, and the fitted values of D and D^ differed insignificantly. For the confined PEO, we used the same equations as those for the confined n-C36H74, except that Sint free(Q,t)/Sfree(Q) and D were calculated from the Rouse model;13,31 specifically, D = kBT/Nξ0, where ξ0 is the monomeric friction coefficient. Free fit parameters were ξ0, RD, and B(Q). Fits are shown in Figure 2. Results are: ξ0 [10-20 kg/ns] = 0.2 ( 0.05 and RD = 20.7 ( 9.7 Å. For the bulk PEO 2k, ξ0 is 0.265 [10-20 kg/ ns];13,21 after we fixed ξ0 at this value and repeated the fit, we obtained RD = 16.4 ( 4.2 Å. From this RD value and Rh = 6 nm, the thickness of the adsorbed layer is determined to be 4.4 nm. For comparison, RF is 4 nm and the hwhm of the pore radius distribution is 0.9 nm. Therefore, the RD value supports the presence of the adsorbed monolayer, just as was found for PEO 3k.13 Note that for PEO, eq 1 is not applicable because of a negligible exchange between the free and adsorbed states. To check whether our spectra allow us to discriminate between an unconfined and confined dynamics, we tried to describe 0 orientation spectra by a bulk polymer (or bulk alkane) model with a Q- and T-dependent background. The fit quality was always worse, as indicated by a significantly higher (at least 5%) χ2 value. Because at 0 orientation only the motions directed perpendicularly to the pore axis are reflected in the spectra, the results we obtained are independent of molecular motions (including the CM diffusion) directed along the pore. Also, note that for the alkane the assumption that the observation time is long enough for the exchange between the states to happen at least several times is supported by the comparison of the traveled distance ((6Dbulkτobsd)0.5) to the pore size. In conclusion, we performed an NSE experiment on melts of n-C36H74 and PEO 2 kg/mol confined in porous silicon and analyzed the spectra in the frame of a two-state model where molecules adsorbed on the surface of the pore are immobile,

MATERIALS AND METHODS

)

)

n-C36H74 and n-C36D74 (Sigma-Aldrich, 99% pure) were used without further purification. Protonated (Mw = 2.1 kg/ mol, Mw/Mn = 1.05) and deuterated PEO (Mw = 2.3 kg/mol, Mw/Mn = 1.05) were synthesized by anionic polymerization in our laboratory. Porous silicon wafers were prepared as described elsewhere.32 The pore space consisted of an array of parallel-aligned channels of roughly circular cross-section and ∼0.3 mm length. The wafers filled with n-C36H74 were freestanding membranes having a porosity (Φ) of ∼0.5, the mean pore radius (Rh) of 5 nm, and the hwhm of the radius distribution of 1.2 nm; the pore surface was covered with a 1 to 2 nm thick native SiO2 layer. The wafers filled with PEO (Φ = 0.64, Rh = 6 nm, hwhm = 0.9 nm) had a nonporous support layer, their total thickness was 0.53 mm, and the pore surface was covered by a thicker SiO2 layer due to the treatment with H2O2. The pore radii were determined from volumetric N2 or Ar sorption isotherms32,33 and had a Lorentzian distribution. In the glovebox flushed with N2, the powder mixtures of deuterated and nondeuterated material (H/D 40:60) were spread on the top of the respective wafers, which were then kept on a hot plate until the pores were completely filled. (To match the coherent SLD of a melt to that of the Si matrix and thus to suppress the elastic scattering, the mass ratio H/D should be ∼80:20.13 For small molecules and for such a high H/D ratio, the corresponding low coherent single molecule scattering intensity together with a substantial spin-incoherent scattering contribution would lead to a low NSE signal. The mass ratio H/D 40:60 results in a nearly maximal single molecule coherent scattering intensity and a relatively low spin-incoherent scattering contribution.) In the NSE technique, the polarization (P) of the neutrons scattered by the sample is recorded as a function of neutron wave vector transfer (Q, Q = 4π/λ sin(θ/2), where θ is the scattering angle and λ is the neutron wavelength) and Fourier time (t). The normalized polarization, P(Q,t)/P(Q), is equal to (I(Q,t) - Iinc(Q,t)/3)/(I(Q) - Iinc(Q)/3), where I(Q,t) and Iinc(Q,t) are the coherent and incoherent intermediate scattering functions, respectively. The factor -1/3 accounts for the change of the polarization in case of the spin-incoherent scattering. In the low-Q region studied here, the contribution


3119


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from the intermolecular coherent scattering is negligible relative to the single molecule coherent scattering that originates from the difference in the scattering lengths of H and D and relative to the spin-incoherent scattering. Therefore, P(Q,t)/P(Q) is just the dynamic structure factor of the molecule, S(Q,t), normalized to its static structure factor, S(Q). If the center-of-mass (cm) and internal (int) dynamics are separated, then S(Q,t)/S(Q) = Scm(Q,t), where Sint(Q,t)/S(Q) = int int int (Sint coh(Q,t) - Sinc(Q,t)/3)/(Scoh(Q) - Sinc(Q)/3). For n-C36H74, we used eq 4 to account for both coherent and incoherent scattering contributions to Sint(Q,t)/S(Q). For PEO 2k, Sint(Q,t)/S(Q) was calculated from equation 11 in ref 13. The experiment was carried out with J-NSE (FRM II, Munich, Germany) at λ = 8 Å, Fourier time ranged from 0.1 to 20 ns, and sample orientation was such that Q was perpendicular to the long axis of the pore;13 that is, Q = Q^. (We call it 0 orientation.) Seven n-C36H74-filled wafers as well as bulk n-C36H74 (H/D 40:60, 1 mm thick) were put in flat sample cells and measured at 364, 435, and 512 K for Q = 0.05, 0.08, 0.11, and 0.15 Å-1. Six PEO-filled wafers were measured at 413 K for the same Q values. For n-C36H74-filled wafers at 512 K and Q = 0.08 Å-1, one spectrum was also recorded at 48 sample orientation (when Q^ ≈ Q ). The correction for resolution was done using the spectra of the activated-carbon powder. We did not correct for the scattering by the sample cell. (This scattering is elastic and taken into account by B(Q) in fits to the spectra of confined n-C36H74 and PEO; it is negligible in the case of the bulk n-C36H74, as indicated by the decay of S(Q,t)/S(Q) to ∼0 (Figure 1).)

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AUTHOR INFORMATION

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Corresponding Author: *To whom correspondence should be addressed. E-mail: a.kusmin@ fz-juelich.de (A.K.); [email protected] (P.H.).

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Present Addresses: z

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA.

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ACKNOWLEDGMENT This work has been supported by the

German Research Foundation (DFG) within the priority program SP 1164, Nano- and Microfluidics, grant no. Hu 850/2.

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Evidence of a Sticky Boundary Layer in Nanochannels: A Neutron

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