Crossover from Normal to Inverse Temperature Dependence in the

J. Phys. Chem. C 2007, 111, 16045-16054

16045

Crossover from Normal to Inverse Temperature Dependence in the Adsorption of Nonionic Surfactants at Hydrophilic Surfaces and Pore Walls† Oliver Dietsch,‡ Anton Eltekov,‡ Henry Bock,§ Keith E. Gubbins,| and Gerhard H. Findenegg*,‡ Institut fu¨r Chemie, Stranski Laboratorium fu¨r Physikalische und Theoretische Chemie, Technische UniVersita¨t Berlin, D-10623 Berlin, Germany, School of Engineering and Physical Sciences, Heriot-Watt UniVersity, Edinburgh EH14 4AS, U.K., and Department of Chemical & Biomolecular Engineering, North Carolina State UniVersity, Raleigh, North Carolina 27695 ReceiVed: June 19, 2007; In Final Form: August 10, 2007

The adsorption of the nonionic surfactant C8E4 from its aqueous solutions to the pore walls of four controlledpore silica glass (CPG) materials of different mean pore widths (10-50 nm) has been studied in a temperature range from 5 to 45 °C, that is, close to the lower critical temperature of liquid-liquid phase separation of the bulk system (Tc ≈ 40 °C). Pronounced S-shaped isotherms, with a normal temperature dependence of the adsorption in the initial low-affinity region but an inverse temperature dependence in the plateau region, are found with all CPG materials. The experimental adsorption isotherms are compared with predictions of a theoretical model (Phys. ReV. Lett. 2004, 92, 135701), which takes into account H-bonding and micelle formation in the bulk and at the surface. It is found that this model reproduces all peculiarities of the adsorption in the present systems. The following conclusions emerge from this analysis: (1) At low bulk concentrations, the surfactant is adsorbed only in monomeric form (Henry’s law behavior). In this regime, there is an energetic driving force for surfactant adsorption (in spite of the fact that nonselective pore walls are assumed), in agreement with the observed normal temperature dependence of the adsorption in this regime. (2) Surface aggregation observed at higher concentrations involves an energy penalty due to the loss of H bonds, which is overcompensated by the gain of (rotational) entropy of the H-bonding sites of the surfactant heads. Accordingly, surface aggregation is entropy-driven and endothermic and thus shows inverse temperature dependence. Hence, the model accounts for the observed crossover in the temperature dependence of the adsorption at the critical surface aggregation concentration.

1. Introduction The adsorption of nonionic surfactants from aqueous solutions onto hydrophilic surfaces like silica has been studied extensively in the past in view of its relevance for wetting, detergency, and related phenomena. A better understanding of surfactant adsorption in narrow pores is of importance for the optimization of surfactant-aided separation processes such as micelleenhanced ultrafiltration or surfactant-aided size-exclusion chromatography. It is commonly believed that the adsorption of surfactants at hydrophilic surfaces involves an aggregation of the molecules. Information about the strength of interaction of surfactant headgroups with the surface and of the hydrophobic interaction among the surfactant tails can be derived from adsorption isotherms and calorimetric adsorption studies.1-6 For surfactant layers on atomically flat surfaces, the amount adsorbed can be determined in situ by ellipsometry7,8 and the thickness of the layer by neutron reflectometry,9,10 while information about the lateral structure of the adsorbed layer can be obtained by grazing †

Part of the “Keith E. Gubbins Festschrift”. * To whom correspondence should be addressed. E-mail: findenegg@ chem.tu-berlin.de. ‡ Technische Universita ¨ t Berlin. § Heriot-Watt University. | North Carolina State University.

incidence small-angle neutron scattering (GISANS)11 and atomic force microscopy (AFM).12 In favorable cases, AFM can provide direct information about the type of surface aggregates prevailing in the plateau region of the adsorption isotherm. For poly(oxyethylene) surfactants (abbreviated as CnEm, where E denotes an oxyethylene group) on hydrophilic silica, it was found12 that aggregate structures similar to those in the bulk solution above the critical micelle concentration (cmc) are formed at the surface. Specifically, surfactants characterized by a large value of the packing parameter13 (such as C10E5 or C10E6) were found to form globular surface aggregates, while surfactants of lower packing parameter (C12E5 and C14E6) may form continuous or fragmented bilayer structures at the surface. The formation of small surface micelles was also reported for the adsorption of a technical-grade nonionic surfactant (Triton X-100) on small colloidal silica beads.14 On the other hand, systematic studies of the adsorption of surfactants in pores and specific effects caused by confinement of the adsorption space are scarce. For the adsorption of nonionic surfactants in hydrophilic porous silica, it was established2,15 that the sigmoidal shape of the isotherm is retained and that the plateau value of the adsorption per unit area decreases with decreasing pore size, in agreement with the predictions of a thermodynamic analysis of the adsorption of nonionic surfactants in cylindrical pores.16 Information about the aggregate structure and dynamics of the surfactant C12E5 in two controlled-pore glass materials of different mean

10.1021/jp0747656 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/05/2007

16046 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Dietsch et al. TABLE 1: Characterization of the CPG Materials by Nitrogen Adsorption: Specific Surface Area as, Specific Pore Volume Wp, and Pore Diameter dp material

as/m2g-1

Vp/cm3g-1

dp / nm

dpa/nm

CPG-75 CPG-170 CPG-240 CPG-500

170 118 91 69a

0.58 0.87 1.06 1.5a

10.3 22.4 35.0 (66)

7.7 15.6 24.2 50

a

Figure 1. Phase diagram of the C8E4 + water system (temperature vs surfactant mass fraction in weight percent) showing the liquid-liquid coexistence curve (ll ) with the lower critical point (LCP)18 and the locus of critical micelle concentrations (cmc).5,19 The horizontal dashed lines indicate the temperatures and the concentration range of the adsorption isotherms measured in this work.

pore sizes was obtained by deuterium MNR. These studies suggested a patchy bilayer structure of the surfactant at the pore walls.17 A characteristic feature of aqueous systems of CnEm surfactants is the occurrence of liquid-liquid immiscibility above some lower critical solution temperature (LCT) on the waterrich side of the phase diagram.18 For this class of surfactants, the cmc decreases with increasing temperature, and at temperatures well above the LCT the cmc line nearly coincides with the phase boundary curve of liquid-liquid coexistence. For the C8E4 + water system, this is illustrated in Figure 1.18,19 A phenomenon closely related to this phase behavior is the anomalous temperature dependence of the adsorption of CnEm surfactants at hydrophilic surfaces, that is, the amount adsorbed in the plateau region of the isotherm increases with temperature.3,4,20 This behavior is attributed to a gradual dehydration of the surfactant heads (i.e., breaking of water-surfactant H bonds) with increasing temperature, which renders the surfactant molecules less-hydrophilic in water and thus favors phase separation and accumulation of the surfactant at the interface. A theoretical model to account for these phenomena was developed recently by two of the present authors.21 It was shown that the anomalous temperature effects arise from strongly orientation-dependent H-bond interactions between water and the surfactant molecules and that the sudden increase in rotational entropy due to the breaking of these bonds leads to the observed anomalous temperature dependence. In this paper, we report a systematic study of the temperature dependence of the adsorption of the surfactant C8E4 in controlledpore glass (CPG) materials with mean pore sizes that range from 10 to 50 nm. CPG silica glasses, like the structurally similar Vycor glasses, are prepared by a spinodal decomposition phaseseparation process, followed by removal of one phase by acid leaching.22,23 This process leads to an interconnected pore system of complex pore shapes and bifurcation patterns but a relatively low degree of surface irregularity (surface fractal dimension D ) 2.20).24 CPG materials have a porosity in the 60-70% range. Their pore structure can be visualized by three-dimensional reconstruction methods.25,26 These materials are well-suited to study the effects of confined geometry on the properties of simple and complex liquids because CPG samples covering a wide range of mean pore widths, yet similar surface properties and pore geometry, are available. The focus of the present paper is, however, on the temperature dependence of the adsorption and on a comparison of the experimental data with the predictions of the theoretical model presented in ref 21.

Values given by the manufacturer.

Adsorption isotherms of C8E4 in the CPG materials were measured in the temperature range from 5 to 45 °C, which covers the region below and just above the LCT of the bulk system (Tc ) 40.7 °C). The temperatures and concentration ranges of adsorption isotherms covered in the present work are indicated by dashed lines in Figure 1. Preliminary results for the adsorption of C8E4 in one of the CPG materials have been presented elsewhere.20,21 The paper is organized as follows. Section 2 gives details of the porosity of the CPG materials and describes the methods employed in the determination of the adsorption isotherms. Section 3 presents the experimental adsorption isotherms and an analysis of the data in terms of a phenomenological model isotherm equation, which allows us to express the temperature and pore-size dependence of the isotherms by a small number of model parameters. The application of the lattice gas theory for the surfactant + water system21 and results for the adsorption in a confined geometry are presented in Section 4. Finally, Section 5 contains a discussion of the experimental results and predictions of the theoretical model and concluding remarks. 2. Materials and Methods 2.1. Materials. Controlled-pore glass (“CPG-10”) materials by Electro-Nucleonics (NJ) were received from Fluka (Germany). Four materials of different mean pore sizes (designated here as CPG-75, CPG-170, CPG-240, and CPG-500, according to their nominal pore size in angstroms) were used in this study. The materials were characterized by nitrogen adsorption at 77 K using an ASAP 2010 volumetric gas adsorption analyzer (Micromeritics). The nitrogen adsorption isotherms exhibit a sharp pore condensation step, a H1-type hysteresis loop and a flat plateau above the pore condensation loop. Values of the specific surface area as (BET multipoint surface area),27 specific pore volume Vp (BJH desorption cumulative pore volume),27,28 and mean pore diameter dp (BJH desorption average pore diameter)27,28 as derived from the sorption isotherms are summarized in Table 1. For CPG-75, the BET surface area agrees with the BJH adsorption cumulative area, whereas for CPG-170 and CPG-240 the BET surface area is intermediate between the BJH adsorption and desorption cumulative areas. For these three materials (CPG-75, CPG-170 and CPG-240), our value of dp is greater than that given by the manufacturer (dp* in Table 1), and we find dp/dH ) 0.759 ( 0.002, where dH ) 4Vp/as is the hydraulic diameter of a cylinder. Values of dp/ dH < 1 are to be expected for materials constituting an interconneted system of cylindrical pores. For the CPG-500 material, for which no nitrogen adsorption data have been obtained in our laboratory, the specific surface area and pore width given by the manufacturer have been used in this work (Table 1). A larger value of the pore width (dp ) 66 nm) would result from the given values of the specific surface area and pore volume on the assumption that dp/dH ) 0.759, as for the other three materials. The surfactant tetra(oxyethylene) monooctylether (C8E4, purity >98%) by Nikko Chemicals was received from Fluka

Normal to Inverse Temperature Dependence

J. Phys. Chem. C, Vol. 111, No. 43, 2007 16047

Figure 2. Chromatographic apparatus used for the surfactant adsorption measurements by frontal analysis (a) and the circulation method (b): L (L1, L2), liquid reservoirs; D, degasser; P (P1, P2), HPLC micropumps; M, mixer unit; COL, chromatographic column; RID, refractive index detector; FM, liquid flow-meter.

Chemicals and used without further purification. The samples were kept under nitrogen gas at 0 °C before use. Deionized water was purified in a Milli-Q water purification apparatus. 2.2. Methods. Adsorption isotherms of the surfactant C8E4 in the four CPG materials were measured by frontal analysis (FA) liquid adsorption chromatography29,30 and by a liquid circulation (LC) technique.31 In the FA experiments, a concentration step from c′ to c′′ is applied at time t ) 0, and the breakthrough curve c(t) of this concentration change at the column outlet is recorded at constant volume flow-rate F. The measurement yields the retention time, tR, of the concentration step in the column given by

tR )

1 (c′′ - c′)

∫0∞ (c′′ - c(t))dt

(1)

The change in the reduced surface excess concentration of the surfactant, Γ, in the adsorption column caused by this concentration step is related to the retention volume VR ) FtR by

c′′ - c′ (VR - VM) Γ′′ - Γ′ ) ms as

(2)

where Γ′′ and Γ′ represent the values of the surface concentration at the bulk concentrations c′′ and c′, respectively, ms is the mass of adsorbent in the column, as its specific surface area, and VM is the dead space (hold-up volume) of the adsorption column. Hence, the adsorption isotherm can be determined in a cumulative way by measuring the retention volumes of a series of concentration steps, starting from pure solvent (c ) 0). The chromatographic apparatus for the FA measurements (Figure 2a) consists of two liquid reservoirs (L1, L2) and HPLC micropumps (P1, P2), on-line degasser (D), the thermostated adsorption column (COL) containing a known mass (ca. 0.5 g) of the CPG material, a differential refractive index detector (RID), and a liquid flow-meter (FM). Solutions of different surfactant concentrations, c, were generated in the mixer unit (M) by mixing a stock solution (supply line 1) with pure water (supply line 2). This was achieved by increasing the volume flow-rate of P1 and simultaneously decreasing that of P2 in steps of 0.1 mL/min, in order to keep the overall flow through the column constant (F ) 1 mL/min). The column was first equilibrated with pure water from L2, and the dead volume, VM, was determined from the elution time of small pulses of D2O placed in the liquid reservoir L1 instead of the stock solution. To measure points of the adsorption isotherm, the

breakthrough curve, c(t), of concentration steps was recorded until the RID signal had reached a constant value within a small error margin. At concentrations below the csac, the retention volume, VR, of the concentration steps was not much greater than the dead volume, VM; hence, great care had to be taken to measure VM and the retention volumes under the same experimental conditions. At concentrations above the csac, the FA technique could not be used because of the large retention times of the concentration steps (several hours) in that regime. In the circulation technique, a known amount of solution of known initial composition is pumped in a closed circuit over the adsorption column until the adsorption equilibrium is established, and the surface concentration Γ is determined from the relation

Γ)

n0(x0 - x) ms as

(3)

where n0 is the amount of solution of initial composition (mole fraction of surfactant) x0, and x is the respective composition of the solution after equilibration with the adsorbent in the column (mass ms, specific surface area as). The experimental setup (Figure 2b) consists of a liquid reservoir (L), degasser (D), a micropump (P), the thermostated adsorption column (COL), and a differential refractive index detector (RID). The system contains ca. 3 g of the adsorbent and 40 mL of liquid solution. Changes in concentration due to adsorption were determined from the refractive index changes by means of a calibration curve. For a given initial solution (n0,x0), the temperature dependence of the adsorption was measured by determining the equilibrium concentration x(T) at different temperatures T. Further points along the adsorption isotherms were obtained by replacing part of the solution in the liquid reservoir by a solution of higher concentration and recalculating (n0,x0). 3. Experimental Results Figure 3 shows adsorption isotherms of C8E4 in the four CPG materials at three temperatures (5, 25, and 45 °C) in a concentration range extending to about c ) 1.5cmc. As indicated in Figure 1, the isotherms at 45 °C represent a temperature above Tc (40.7 °C) and extend almost to the two-phase coexistence curve of the C8E4 + water system, while the isotherms at the two lower temperatures are well in the range of complete miscibility. Qualitatively, the adsorption isotherms in the four porous glass materials exhibit a similar behavior. Up to a concentration c ≈ 0.7cmc, very low adsorption is observed, followed by a region in which the surface concentration increases sharply, and finally the region above the cmc in which the surface concentration exhibits only a weak further increase with increasing bulk concentration. The steep increase of Γ in the intermediate regime is attributed to surface aggregation. A critical surface aggregation concentration (csac) can be defined as the onset concentration of the surface aggregation. We first analyze the behavior in the low-affinity region below the csac and then turn to the aggregative adsorption above the csac. 3.1. Low-Affinity Region. Figure 4a shows the lowconcentration (low-affinity) region of the isotherms of C8E4 in CPG-170 for the three experimental temperatures. In this region, the isotherms exhibit Langmuir-type behavior and a normal (negative) temperature dependence of the adsorption, but strong deviations from Langmuir behavior appear above c ≈ 5 mM for the 45 °C isotherm, and above c ≈ 6 mM for the 25 °C isotherm, indicating the onset of aggregative adsorption. (For


Dietsch et al.

Figure 3. Adsorption isotherms of C8E4 in the controlled-pore glass materials CPG-75 (a), CPG-170 (b), CPG-240 (c), and CPG-500 (d): Experimental data for 5 °C (9), 25 °C (b), and 45 °C (2) and fit of the data by eq 5 (full lines). Values of the fit parameters are given in Table 2.

the 5 °C isotherm, the onset of surface aggregation is at a concentration c ≈ 10 mM, i.e., beyond the concentration range shown in the figure.) The Henry’s law adsorption constant KH ) limcfo(Γ/c) was determined by fitting a Langmuir relation to the low-affinity region of the isotherms (below the onset concentration). Values of KH determined in this way are given in Table 2.The uncertainty in KH is rather large ((10%) because of the small number of data points in the low-affinity region, which makes it difficult to determine the two parameters of the Langmuir equation independently. The mean molar enthalpy of adsorption in the low-affinity region, ∆adsh0, was obtained from the temperature dependence of KH by the thermodynamic relation

∆adsh0 ) -R

d ln KH d(1/T)

(4)

Within experimental accuracy, the data can be represented by a linear relation of ln(KH) versus 1/T. As a typical example, Figure 4b shows the quality of this fit for C8E4 in CPG-170. Values of ∆adsh0 and KH(298) for C8E4 in the four CPG materials are given in Table 3. The values of KH (298) derived from the linear regression show some increase with increasing pore size, similar to the trend in the original data of KH of Table 2. For ∆adsh0, this analysis yields values of about -25 kJ mol-1, independent of pore size within the limits of experimental error, which are estimated to be (20%. 3.2. Aggregative Adsorption Region. The entire adsorption isotherms can be represented by the two-step adsorption model by Zhu and Gu32,33 in which the first step involves the adsorption of individual surfactant molecules at surface sites, and the second

Figure 4. (a) Initial region of the adsorption isotherms of C8E4 in CPG-170 shown on an expanded scale of Γ: Experimental data (9, 5 °C; b, 25 °C; 2,45 °C) and initial slope of the isotherms (full lines) as derived by fitting the Langmuir equation to the data below the onset concentration; (b) Henry’s law constant KH of C8E4 in CPG-170 at the three experimental temperatures in a ln(KH) vs 1/T representation; the enthalpy of adsorption ∆adsh0 for the Henry’s law region is derived by linear regression of these data (full line).



TABLE 2: Experimental Values of the Henry’s Law Constant KH, and Best-Fit Values of the Parameters c0, Γmax, and N of Eq 6 for the Adsorption Isotherms of C8E4 at the Silica/Water Interface of the Controlled-Pore Glass Materials at Three Experimental Temperatures, T materials CPG-75 CPG-170 CPG-240 CPG-500

T, °C

KH, 10-9 m

c0, 10-3 mol L-1

Γmax, 10-6 mol m-2

N

5 25 45 5 25 45 5 25 45 5 25 45

13.9 7.8 3.6 16.8 10.0 4.0 30.1 14.3 4.1 25.2 12.7 6.0

9.52 5.74 4.22 10.01 6.41 5.15 8.49 5.97 4.79 8.69 5.87 4.59

2.20 2.44 2.94 3.85 4.68 6.13 5.54 6.04 6.87 5.82 6.18 6.85

8.61 8.38 13.02 12.46 12.49 24.41 9.98 11.29 26.79 12.54 13.09 18.29

TABLE 3: Henry’s Law Constant KH at 298 K and Mean Molar Heat of Adsorption in the Low-Affinity Region, ∆ads h0, for the Adsorption of C8E4 in the CPG-10 Materials: Results of the Linear Regression of the Experimental Values of KH of Table 2 on the Basis of Equation 4 material

KH(298), 10-9 m

-∆ads h0, kJ mol-1

CPG-75 CPG-170 CPG-240 CPG-500

7.1 ( 0.7 8.5 ( 1.0 11.6 ( 1.2 12.1 ( 1.2

25 ( 5 26 ( 5 36 ( 5 26 ( 5

step arises from aggregation of further surfactant molecules at these “anchor molecules”. This model yields the isotherm equation

Γ ) Γm

(N1 + k c )

k1c

N-1

2

1 + k1c(1 + k2cN-1)

(5)

where Γm is the limiting value of the adsorption at high concentrations, c, of the solution, k1 and k2 are the equilibrium constants of the first and second adsorption step, and N is the aggregation number of the surface aggregates. Equation 5 can be expressed in terms of Henry’s law constant, KH, and a reduced concentration, x ) c/c0, as

Γ)

KHc0 x + (Γm/N)xN 1 + NKHc0 x/Γm + xN/N

(6)

where c0 ) (Nk1k2)-1/N and KH ) k1Γm/N. For concentrations x , 1, eq 6 reduces to a Langmuir equation for the adsorption of single surfactant molecules, with a limiting surface concentration Γm/N, but at x > 1 the term xN causes a steep increase of the isotherm that leads to a limiting surface concentration, Γm, characteristic of the surface aggregates of N molecules. Hence, c0 represents the critical surface aggregation concentration (csac), defined operationally for N . 1 as the concentration in the steeply ascending region of the isotherm at which Γ reaches a value Γm/N. Equation 5 was fitted to the adsorption isotherms in order to parametrize the observed dependence of the surface aggregation concentration, c0, and the limiting surface concentration, Γm, on temperature and pore width of the matrix. In this analysis, the values of the Henry’s law constant, KH, obtained from the low-affinity region of the adsorption isotherm were taken as input parameter for k1 ) NKH/Γm and the quantities Γm, k2, and

N were obtained by nonlinear regression. Resulting values of the parameters Γm, N, and c0 ) (Nk1k2)-1/N are collected in Table 2, together with those of KH, obtained from the analysis of the low-affinity region (Section 3.1). From a parameter study, we estimate the uncertainty in Γm to (2% and the uncertainty in c0 and N to about (5%. The fit of the adsorption data by this model is shown by the full curves in Figure 3a-d. As can be seen, the model gives a good representation of the steep increase of the isotherms in the regime of the aggregative adsorption and of the plateau at 5 and 25 °C. However, the model does not reproduce the weak but systematic increase of the adsorption in the plateau region observed at 45 °C (and 25 °C for the material with the widest pores). For the present systems, the parameters Γm, c0, and N reveal the following trends: (i) The limiting adsorption, Γm, increases with temperature for each of the four CPG materials. In all cases, the increment in Γm for the 25-45 °C range is greater than that for the 5 to 25 °C range, indicating that the inverse temperature dependence of the aggregative adsorption becomes more pronounced as the upper miscibility gap of the bulk system is approached. High values of Γm are found for CPG-500 and CPG-240. The highest value, Γm ≈ 6.9 µmol m-2 (for CPG-500 and CPG-240 at 45 °C), corresponds to an apparent cross-sectional area of 0.24 nm2 per molecule in a hypothetical monolayer arrangement (which is physically implausible), or an area of 0.48 nm2 per molecule in a bilayer. On the assumption that confined geometry and curvature effects of the pore wall are negligible in these two CPG materials, the tetra(oxyethylene) headgroups are expected to occupy an area of 0.36 nm2 in a saturated bilayer.5,34 Hence, the highest value of Γm implies a fragmented bilayer structure occupying about 75% of the available area of the pore wall. The lower values of Γm observed in these materials at 25 and 5 °C then indicate that the fraction of uncovered surface increases as the temperature is lowered. By the same argument, the low values of Γm for C8E4 in CPG-75 (2.2 µmol m-2 at 5 °C) then correspond to a patchy bilayer occupying only about 25% of the pore wall. In this case, it is probably more realistic to assume quasi-spherical surface aggregates. In any case, the drastic decrease of the nominal surface coverage from CPG500 to CPG-75 must be attributed to confinement in the pores, because the pore diameter of CPG-75 (∼10 nm) is not much greater than the width of two bilayers of C8E4. (ii) For each of the four CPG materials, the critical surface aggregation concentration, c0, decreases with increasing temperature. Because surface aggregation is expected to be similar to micelle formation in the bulk, it is of interest to look at changes in c0 relative to the bulk cmc, by considering the parameter c0/cmc and its dependence on temperature and pore size. Such a graph is shown in Figure 5. It is based on the following values of the cmc (mM):5,19 11.8 (5 °C), 7.5 (25 °C), and 5.8 (45 °C), corresponding to a mean enthalpy of micellization of 13 kJ mol-1, in reasonable agreement with the values of 15.8 kJ mol-1 5 and 16.8 kJ mol-1 35 obtained by titration calorimetry. Figure 5 shows that most values of c0/cmc are between 0.72 and 0.82, except for CPG-170, for which somewhat higher values are found (0.85 to 0.88). For three of the four CPG materials, c0/cmc exhibits a weak increase with temperature. The deviations in the values of c0/cmc for CPG170, and in its temperature dependence in the case of CPG-75, are likely to reflect experimental artefacts rather than genuine effects. (iii) The parameter N has values typically in the range 10 to 20 for C8E4 in the present systems. For each CPG material, similar values of N are found for the two lower temperatures


Dietsch et al. TABLE 4: System Parameters for the Theoretical Adsorption Model Adopted in the Present Study vdW WW ) 1.0 hb WW ) 5.4 q ) 106 φh ) 8.0

vdW WS ) 0.301 hb WW ) 5.4

vdW SS ) 1.0 hb WW ) 0.0

c1 ) -2.0

c2 ) -0.003

tension, which is proportional to c1, and a contribution from the head/head repulsion proportional to c2:

µM(l ) ) l 3(µS + φh + c1l Figure 5. Quantity c0/cmc (ratio of critical surface aggregation concentration and bulk cmc at the respective temperature) as a function of temperature T for C8E4 in the four CPG materials: b, CPG-75; 9, CPG-170; 4, CPG-240; [, CPG-500.

but a significantly higher value at 45 °C. This is a signature of a steeper rise of the adsorption isotherm in the surface aggregation regime at this temperature T > Tc. For CPG-75, the values of N are smaller (typically by a factor 2/3) than those in the three materials with wider pores. This finding is in line with the lower values of Γm in this material as compared with the materials with wider pores. It supports the view that surface aggregates of smaller size are formed in this matrix because of the confinement effect. 4. Theoretical Analysis 4.1. Adsorption Model. The model used to study the adsorption behavior of aqueous surfactant solutions has been introduced earlier.21 The starting point is a lattice model of a surfactant + water mixture using a simple cubic lattice with lattice constant l. To describe aqueous solutions of surfactant molecules, we must include hydrogen bonding and micelle formation in the model. To account for H-bonding between pairs of water molecules and between water and surfactant, we follow the general idea of the H-bond model of Walker and Vause.35 A characteristic property of H bonds is that they are highly directional. This results in a very large ratio of the total number of orientational states to bonding states. To model H bonds, a scalar orientation variable σ ) 1,2,...,q is assigned to each molecule. Neighboring molecules i and j form a H bond if σi ) σj. This results in the required high ratio of total to bonding states q2/q. Using a low-temperature expansion of this model, it can be (approximately) mapped onto an equivalent model without direction-dependent interactions. As a result, the explicit H bonds are replaced by a temperature-dependent interaction. This leads to an interaction energy between fluid species ff(T) of the form

ff(T) ) vdW + T ln{q-1/3[exp(hb ff ff /T) - 1] + 1}

(7)

which is the sum of a van der Waals part, vdW ff , and the H-bond part, where T is the temperature and hb is the strength of the H ff bonds. To account for the formation of micelles, we introduce a “superlattice” with a lattice constant that is l ) 2,3,... times larger than the lattice constant l of the original lattice, depending on the size of the micelles. Micelles consisting of l 3 surfactant molecules are now introduced as a third species and placed on the nodes of the superlattice. The chemical potential of the micelles, µM(l ), which is needed because the model is developed in the grand-canonical ensemble, consists of four parts:36,37 the chemical potential of the surfactant molecules, µS, a hydrophobic potential, φh, a contribution from the micelle core/water surface

-1

+ c2l 3)

(8)

Solving the model within the mean-field approximation38 on the level of the superlattice yields an expression for the grandcanonical potential density, ω, from which phase diagrams can be obtained. For a bulk solution, ω is given by

(

ω ) T FM(ln FM - l 3 ln q) + (1 - FM)

{

ln(1 - FM) + l 3

[

{

[

]})

-

]}

(9)

FW ln(FWq-1) + FS ln(FSq-1) + (1 - FW - FS) ln(1 - FW - FS)

FM µM + 3F2M l 2SS+ (1 - FM) l 2

6FM(FWSW + FSSS) + l (FW µW + FSµS) + 3(l - FM)(FW2WW + FWFSSW + FS2SS)

In eq 9, FM denotes the number density of micelles on the superlattice, FS and FW are the number densities of surfactant and water on sites of the original lattice not occupied by micelles, and µS and µW are the chemical potentials of surfactant and water. For clarity, we have omitted to indicate the explicit temperature dependence of the interaction parameters WW and SW. An equivalent expression for the grand-canonical potential density of the confined system can be obtained by summing all contributions of ω over the layers of the superlattice. The resulting expression for the confined system is similar in spirit to eq 9 but consisting of a considerably larger number of terms. For this reason, the full expression is not given here. We obtain thermodynamic information from thermodynamically stable states, which can be found by minimizing ω with respect to the densities and l. Values of the system parameters used in this study are listed in Table 4. The particular choice of the parameters is explained in ref 21. For the following discussion, it is important to keep in mind that the average van der Waals interaction between surfactant and water molecules is much smaller than the arithmetic mean of the interaction between pairs of water and pairs of surfactant molecules, leading to a strong hydrophobic effect. H bonds can be formed between pairs of water molecules and between water and surfactant molecules. For simplicity, we have assumed that these two types of H bonds are equally strong. Furthermore, the values of c1 and c2 of eq 8 were chosen such as to yield a micelle size l ) 4 that reasonably approximates the experimental aggregation number (l 3) of the surfactant C8E4. In this paper, we study surfactant adsorption in a slit geometry with a slit width h ) 60l ) 15l (15 micelle diameters), which excludes effects arising from confinement. For given values of the overall surfactant mole fraction in the bulk solution, xbS, the calculations yield values of the local surfactant mole fraction xS(z) in the individual layers z ) 1,2,...,15 of the superlattice.



Note that in all calculations hard walls are assumed, which implies that the pore walls are nonselective. 4.2. Results. The model introduced above is now used to analyze surfactant adsorption from aqueous solution. The adsorbed amount is defined here in the same way as in eq 3 as a reduced surface excess concentration of the surfactant relative to the amount of surfactant in a volume of bulk solution containing the same number of molecules, viz.

Γ)

ν 2

∑z (xS(z) - xbS)

(10)

where xS(z) is the surfactant mole fraction in layer z of the slit pore, xbS is the respective mole fractions in the bulk solution, and ν is the number of surface sites of the superlattice per unit area. The summation is taken over all layers of the superlattice. The factor 1/2 in eq 10 accounts for the presence of two walls in the case of a slit pore. Figure 6 presents adsorption isotherms as graphs of the dimensionless reduced surface excess concentration Γ/ν versus surfactant mole fraction for the temperatures 5, 25, and 45 °C. It can be seen that at low xbS the adsorption is very small, but Γ increases sharply above a certain onset value and approaches a plateau value at higher xbS. This behavior closely resembles that of the experimental adsorption isotherms, where the onset concentration was identified as critical surface aggregation concentration (csac). Figure 6a shows that the plateau value of the adsorption increases with temperature while the onset concentration (csac) decreases with increasing temperature. For the critical micelle concentration of the bulk system the present model gives xcmc ) 0.00434 (5 °C), 0.00367 (25 S °C), and 0.00334 (45 °C), that is, values greater than the onset concentration of surface aggregation (csac) at the respective temperature. Inspection of Figure 6a indicates that xcmc repreS sents a concentration in the region in which the isotherms level off toward the plateau value of Γ. The initial region of the adsorption isotherms is plotted on a highly enlarged scale in Figure 6b to show that in this regime the adsorption exhibits a normal temperature dependence. Figure 6c shows the crossover from the normal to the inverse temperature dependence of the adsorption. All of these features are in qualitative agreement with the experimental isotherms presented in Figures 3 and 4a. Figure 7 shows the dependence of the surfactant mole fraction profile, xS(z), on the bulk mole fraction, xbS, for three different temperatures. At the lowest temperature (Figure 7a), an increase of xbS initially leads only to a slight increase of the surfactant concentration close to the walls. As xbS exceeds the onset value, the surfactant mole fraction in the surface layers (z ) 1 and z ) 15) shows a pronounced increase and reaches fairly high values before the surfactant concentration in the center of the pore shows noticeable changes. The point at which the surfactant concentration in the center of the pore increases noticeably represents the bulk cmc. A similar behavior is found at 25 °C (Figure 7b). At 45 °C (Figure 7c), the mole fraction profiles exhibit a discontinuity of the surfactant concentration in the surface layer at xbS ) 0.00311, indicating a surface phase transition. This observation is in line with the steep rise of the experimental adsorption isotherms at 45 °C in the wide-pore materials. Whether or not this represents a true first-order surface phase transition remains open in view of the fact that lattice models generally overemphasize first-order phase transitions. Note that only in the two outermost layers (z )1 and z ) 15) the mole fraction xS(z) differs significantly from that in the central layer (z ) 8), whereas in all other layers xS(z) is similar to xbS. This implies that the surface excess of the surfactant is

Figure 6. (a) Surfactant adsorption isotherms from the theoretical model, plotted as dimensionless reduced surface excess concentration Γ/ν vs surfactant mole fraction xbS: - 5 °C, ----- 25 °C, - ‚ - 45 °C; note that adsorption at 45 °C is discontinuous. (b) Henry’s law region of the adsorption isotherms indicating the normal temperature dependence of Γ in this regime; (c) adsorption isotherms plotted as ln(Γ/ν) vs xbS to show the crossover from normal to inverse temperature dependence of the adsorption at the onset of aggregative adsorption.

due almost entirely to surfactant in the outermost layer of the superlattice. At the highest temperature (45 °C), the local mole fraction in these two layers jumps from low values to nearly 1 at the surface phase transition. On the basis of eq 10, this leads to a reduced adsorption Γ/ν = 1 above the phase transition (Figure 6a), which implies that the surfaces are completely covered by a monolayer of micelles (see below). In the pore as well as in the bulk solution, the surfactant can exist either in monomeric form (“free” surfactant solubilized in water) or in micellar form. Figure 8 shows the mole fractions of free surfactant and surfactant in micellar aggregates as a function of the bulk mole fraction. At low xbS, only free surfactant molecules are found in the confined solution. Their concentration first increases linearly with xbS but levels off in the region of the csac, when the concentration of surfactant bound in micellar aggregates rises steeply. Because the csac is


Dietsch et al.

Figure 8. Mole fraction of free surfactant and surfactant molecules aggregated to micelles as a function of the bulk mole fraction of surfactant xbS for three temperatures: - 5 °C, ----- 25 °C, and - ‚ 45 °C.

If the hydrophobic effect is the driving force for adsorption, then surfactant adsorption is expected to be an exothermic process and thus should show normal temperature dependence. A normal temperature dependence is indeed predicted by our theory for concentrations below the csac (Figure 6b and c). At these low concentrations, the surfactant exists only in monomeric form (cf. Figure 8) and thus adsorption should follow Henry’s law with linear adsorption isotherms. Figure 6b indicates that Henry’s law is indeed valid for surfactant adsorption well below the csac. Hence, the present theory reproduces not only the inverse temperature dependence in the regime of the aggregative adsorption but also the normal temperature dependence of the adsorption in the Henry’s law region. The crossover of the temperature dependence of surfactant adsorption from normal to inverse can be seen in Figure 6c. As expected from the discussion above, the crossover takes place at the csac where the first adsorbed aggregates are formed in the confined solution. 5. Discussion and Conclusions

Figure 7. Surfactant concentration profiles across the slit pore for a series of bulk mole fractions xbS at three temperatures: (a) 5 °C, (b) 25 °C, and (c) 45 °C. The concentration profiles are expressed by the local mole fraction xS(z) in the individual layers of the superlattice; the outermost layers (z ) 1 and z ) 15) are in contact with the pore walls. The slit width corresponds to 15 micelle diameters.

smaller than the cmc at which micelle formation in the core of the pore commences, the micelles in the confined system must be adsorbed on the pore walls. In other words, the increase of xS(z) in the two outermost layers of the superlattice at bulk mole fractions near the csac exhibited in Figure 7 must be due to surface micelle formation. At bulk mole fractions below the csac, only free surfactant molecules are found in the confined solution. Because in our model the pore walls are nonselective (i.e., not preferentially adsorbing either surfactant or water), the driving force of adsorption is solely the hydrophobic effect, which favors the reduction of the contact area between water and other species.

The experimental results of this work corroborate earlier findings on the temperature dependence3,4,20 and pore-size dependence2,15 of the adsorption of CmEn-type nonionic surfactants at the surface of hydrophilic silica. By studying the adsorption in a temperature range near the LCP of the surfactant + water system, the close relation between the phase behavior of the bulk system and the aggregative adsorption at hydrophilic surfaces could be demonstrated. Because the adsorption was studied in CPG materials of widely different pore sizes, it was also possible to map out the transition from the adsorption behavior on open surfaces to the situation where adsorption is affected by confinement effects. Qualitatively, the same adsorption behavior is found in all of these materials. All adsorption isotherms exhibit a low-affinity region at low bulk concentrations in which the adsorption is exothermic, and a sharp transition to the regime of the aggregative adsorption still below the cmc of the bulk solution, in which the adsorption is endothermic. For one of our systems (C8E4 in CPG-240), the enthalpy of adsorption derived from the temperature dependence of the adsorption isotherms can be compared with direct calorimetric measurements.5 For the initial low-affinity region (isolated adsorbed molecules), flow calorimetry yields a differential enthalpy of adsorption of -15 kJ mol-1 on average, decreasing in magnitude with increasing surface coverage. Extrapolation of the experimental data to zero surface coverage gives a value of -20 kJ mol-1, in rather good agreement with

Normal to Inverse Temperature Dependence the mean enthalpy of -25 kJ mol-1 derived on the basis of eq 4 for three of our systems (Table 3). For the region of the aggregative adsorption above the csac, flow calorimetry yields a differential enthalpy adsorption of +15 kJ mol-1 (independent of surface coverage), which is nearly equal to the enthalpy of micellization in the bulk (15.8 kJ mol-1) and consistent with the conjecture that aggregative adsorption at the csac is similar to micellization in the bulk solution.5 Adsorption isotherms alone cannot provide conclusive information about the structure of the surface aggregates. The plateau values of the adsorption isotherm (Γm) are compatible with a fragmented bilayer or an assembly of small globular surface micelles. As mentioned in the Introduction, either of these aggregate structures has been observed for CmEn surfactants on atomically flat hydrophilic surfaces, depending on the packing parameter of the surfactant. The deuterium NMR study of the adsorption of C12E5 in CPG-75 and CPG-240 suggested a flat bilayer structure of the surface aggregates at the pore walls for this surfactant.17 The situation may be different, however, for C8E4 in view of its smaller packing parameter, which favors globular micelles. It is possible that initially at the csac isolated globular surface aggregates are formed, and a gradual transition to patchy bilayer structures occurs at higher concentrations. Such a scenario is consistent with a GISANS study of the adsorption of C8E4 on a hydrophilic silicon wafer.11 We may conclude, therefore, that a model of spherical surface micelles as invoked in ref 21 and adopted in this work is physically plausible for this surfactant. Our study shows that the onset concentration of surface aggregation (csac) is linked to the critical micelle concentration of the bulk solution at the respective temperature. Although the cmc of C8E4 changes by about a factor of 2 in the experimental temperature range, the ratio of csac and cmc (expressed as c0/ cmc) changes by no more than 10% in the present systems, which is almost within the combined error limits of the cmc and c0 data. For the present systems, we also observe no systematic dependence of the quantity c0/cmc on the pore size. However, this conclusion holds only for pore sizes >10 nm studied in this work. Studies of surfactant adsorption in silicas of more narrow (yet less well-defined) pores2,15 indicate a decrease of the csac at mean pore widths below 10 nm, in agreement with a theoretical analysis based on a Kelvin-like expression for the pore curvature on the adsorption.1 This conclusion is supported by ongoing studies in our laboratory of the adsorption of C8E4 in ordered mesoporous silica (MCM41 and SBA-15) with pore sizes in the range of 3-8 nm. The present work shows that the model of ref 21, which incorporates micelle formation and H-bonding, qualitatively reproduces all important features of the adsorption of the nonionic surfactant at the hydrophilic surface. The results of the theoretical model are in very good agreement with the experimental observations. Specifically, they justify the Henry’s law treatment at low concentration and provide an explanation of the nature of the two adsorption regimes and the crossover from one to the other. The normal temperature dependence of surfactant adsorption in the Henry’s law region can be understood directly from our model. Consider an aqueous solution containing only a single surfactant molecule confined by nonselective pore walls, and first assume that H-bonding is absent. By transferring the surfactant molecule from the center of the pore to the pore wall, the system gains hydrophobic energy by reducing the area of contact of water with the pore walls and with the surfactant molecule at the same time. (There might also be a contribution

J. Phys. Chem. C, Vol. 111, No. 43, 2007 16053 from forming surfactant/wall contacts.) This process is energydriven (exothermic) and thus will have normal temperature dependence. When H-bonding is now included this should have no influence on the driving force as long as the surfactant exists only in monomeric form. To see this, recollect that our model is based on the following two assumptions: (i) the number of H bonds per contact is the same for water/water and water/ surfactant pairs, and (ii) the two types of H bonds are equally strong. Because of assumption i, the process of transferring the surfactant molecule from the center of the pore to the pore walls does not change the total number of H bonds in the system because the increases in the number of water/water H bonds will be compensated by the loss of an equal number of water/ surfactant H bonds. Because in our model both types of H bonds are energetically equivalent (assumption ii), H-bonding constitutes neither an energetic nor an entropic driving force for adsorption in this regime. Assumption ii may not be strictly valid for all types of nonionic surfactants, but surfactant-specific effects seem to be rather small as indicated by the exothermic character of surfactant adsorption at low concentrations found in the experiments (Figure 4a). This situation changes dramatically when the surfactant concentration is high enough to induce aggregation. The main difference between the adsorption of single molecules and aggregation is that the latter leads to the formation of surfactant/ surfactant contacts that do not show H-bonding. Thus, aggregation leads to an energy penalty due to the loss of H bonds, which is, however, overcompensated by an increase in rotational entropy of the H-bonding sites of the surfactant headgroups. As a consequence, aggregation is driven by the gain of rotational entropy, and thus it represents an endothermic process and shows inverse temperature dependence. In conclusion, the present work shows that a model that takes into account H-bonding and micelle formation can account for all significant features of surfactant adsorption at hydrophilic surfaces and pore walls. In particular, our results show that (1) at low (bulk) surfactant concentrations surfactant is adsorbed only in monomeric form (Henry’s law behavior). (2) There is an energetic driving force for surfactant adsorption in this regime, in spite of the fact that hard (nonselective) pore walls are assumed. This implies that adsorption is exothermic and shows a normal temperature dependence, even in the absence of energetically favorable sites at the surface. (3) Surface aggregation of the surfactant involves an energy penalty due to the loss of H bonds, which is, however, overcompensated by the gain of (rotational) entropy of the H-bonding sites of the surfactant heads. Hence, surface aggregation is entropy-driven and endothermic and thus shows an inverse temperature dependence. (4) The temperature dependence of surfactant adsorption from aqueous solution shows a crossover from normal to inverse at the csac where the first (adsorbed) aggregates are formed in the confined solution. Acknowledgment. We are grateful to P. Klobes for his help in the characterization of the CPG materials, and R. Dabiri for performing some of the surfactant adsorption measurements. This research was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. FI 235/15 and by the NSF under Grant No. IN T-0329695. References and Notes (1) Bo¨hmer, M. R.; Koopal, L. K.; Janssen, R.; Lee, E. M.; Thomas, R. K.; Rennie, A. R. Langmuir 1992, 8, 2228. (2) Giordano, F.; Denoyel, R.; Rouquerol, J. Colloids Surf., A 1993, 71, 293.

16054 J. Phys. Chem. C, Vol. 111, No. 43, 2007 (3) Partyka, S.; Lindheimer, M.; Faucompre, B. Colloids Surf., A 1993, 76, 267. (4) Wittstock, C.; Kohler, H.-H.; Seidel, J. Langmuir 1996, 12, 5557. (5) Kira´ly, Z.; Bo¨rner, R. H. K.; Findenegg, G. H. Langmuir 1997, 13, 3308. (6) Drach, M.; Narkiewicz-Michalek, J.; Rudzinski, W.; Findenegg, G. H.; Kira´ly, Z. Phys. Chem. Chem. Phys. 2002, 4, 2307. (7) Tiberg, F. J. Chem. Soc., Faraday Trans. 2 1996, 92, 531. (8) Matsson, M. K.; Kronberg, B.; Claesson, P. M. Langmuir 2004, 20, 4051. (9) Gilchrist, V. A.; Lu, J. R.; Staples, E.; Garett, P.; Penfold, J. Langmuir 1999, 15, 250. (10) Howse, J. R.; Steitz, R.; Pannek, M.; Simon, P.; Schubert, D. W.; Findenegg, G. H. Phys. Chem. Chem. Phys. 2001, 3, 4044. (11) Steitz, R.; Mu¨ller-Buschbaum, P.; Schemmel, S.; Cubitt, R.; Findenegg, G. H. Europhys. Lett. 2004, 67, 962. (12) Grant, L. M.; Tiberg, F.; Ducker, W. A. J. Phys. Chem. B 1998, 102, 4288; see also Tiberg, F.; Brinck, J.; Grant, L. M. Current Opin. Colloid Interface Sci. 2000, 4, 411. (13) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1 1981, 77, 601. (14) Despert, G.; Oberdisse, J. Langmuir 2003, 19, 7610. (15) Gu, T.; Zhu, B.-Y. Colloids Surf. 1990, 44, 81. (16) Huinink, H. P.; de Kaizer, A.; Leermakers, F. A. M.; Lyklema, J. Langmuir 1997, 13, 6452. (17) Qiao, Y.; Scho¨nhoff, M.; Findenegg, G. H. Langmuir 2003, 19, 6160. (18) Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 182. (19) Pakusch, A. Ph.D. thesis, Universita¨t Go¨ttingen, 1983. (20) Findenegg, G. H.; Braun, C.; Lang P.; Steitz, R. In Supramolecular Structure in Confined Geometry; Manne, S., Warr, G. G., Eds; ACS

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Crossover from Normal to Inverse Temperature Dependence in the

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