Isotherms and Kinetics of Water Vapor Sorption ... - ACS Publications

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Isotherms and Kinetics of Water Vapor Sorption/Desorption for Surface Films of Polyion−Surfactant Ion Complex Salts Charlotte Gustavsson and Lennart Piculell* Physical Chemistry, Department of Chemistry, Lund University, Box 124, SE-22100 Lund, Sweden S Supporting Information *

ABSTRACT: Thin films of “complex salts” (CS = ionic surfactants with polymeric counterions) have recently been shown to respond to humidity changes in ambient air by changing their liquid crystalline structure. We here report isotherms and kinetics of water sorption/desorption for ∼10−100 μm films of alkyltrimethylammonium polyacrylate CS, measured in a dynamic gravimetric vapor sorption instrument over a 0−95% relative humidity (RH) range. The sorption per ion pair was similar to that observed for common ionomers. A kinetic model for the water exchange is presented, assuming that the “external” transport between the vapor reservoir and the film surface is rate-determining. The model predicts that the water content, after a small stepwise change of the reservoir RH, should vary exponentially with time, with a time constant proportional to both the slope of the sorption isotherm and the film thickness. These predictions were confirmed for our films over large RH ranges, and the external mass transfer coefficient in our setup was calculated from the experimental data. Expressions derived for the Biot number (ratio of characteristic times for internal and external water transport) for the considered limiting case strongly indicate that external water transport should quite generally affect, or even dominate, the measured kinetics for similarly thin hydrated films.

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INTRODUCTION Recently, the present authors have shown that films of polyion−surfactant ion “complex salts” (CS) can readily be cast from ethanolic solutions onto a variety of solid surfaces and that these films change their internal liquid crystalline structure in response to changes of the water activity in a surrounding environment (a humid atmosphere or an aqueous solution).1−3 The investigated CS belong to the well-studied family of alkyltrimethylammonium surfactants with polyacrylate counterions, here denoted as CnTAPAp, where n is the number of carbons in the surfactant alkyl tail and p is the degree of polymerization of the polyacrylate. The CnTAPAp complex salts are not soluble in water, but are quite hygroscopic, and have been found to take up on the order of 10−20 wt % water from normally humid indoor air.4−6 Like the analogous common cationic surfactants (CnTACl or CnTABr), the hydrated CnTAPAp complex salts feature a sequence of different liquid crystalline structures with changing water content, as has been shown in detailed studies of bulk CS samples.4−7 Our recent studies demonstrated that the same structural transitions, involving the 3D-hexagonal close-packed micellar, cubic © 2016 American Chemical Society

micellar, 2D-hexagonal, or rectangular phases, can be induced in thin surface films (10−100 μm thick) by suitable changes in the humidity of the air surrounding the film.1 Very recently, the kinetics of the structural response of the film to rapid humidity changes, and their dependence on the identity of the CS, was studied in detailed small-angle X-ray scattering (SAXS) experiments.2 For a better understanding of the “structural responsivity” of CS films, it is important to acquire basic knowledge on both equilibrium and kinetic aspects of the water exchange between a hydrated CS film and a surrounding hydrated gas phase. This is the basic motivation for the present study. To the above ends, we have here used a sensitive dynamic gravimetric vapor sorption (DVS) instrument to obtain water sorption isotherms, and characteristic time constants for water exchanges, following step changes in relative humidity (RH), over the range 0−95% for 10−100 μm thick films of four Received: March 22, 2016 Revised: June 21, 2016 Published: June 21, 2016 6778

DOI: 10.1021/acs.jpcb.6b02983 J. Phys. Chem. B 2016, 120, 6778−6790

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The Journal of Physical Chemistry B different CnTAPAp complex salts. These films were produced using exactly the same protocol as in our previous studies.1−3 The information that these data provide on hydrated complex salt films will be thoroughly discussed. Specifically, for the interpretation of the kinetic data, we present a detailed kinetic model of water exchange under the limiting case where the transport of water from the humid gas reservoir to the surface of the film is the rate-determining process. We show that, in this limiting situation, the kinetic response of a standard sorption/desorption experiment (measuring the water content versus time after a small stepwise change of the relative humidity) should display a number of quite characteristic features. By comparing with experimental data we then indeed find that, under a large part of the investigated experimental conditions, the rate of sorption/desorption for our CS or surfactant films is entirely determined by the water transport in the vapor phase; all other equilibration processes, including the internal equilibration of the water content through diffusion inside the film, are much more rapid. Detailed water sorption isotherms for some simple surfactants have been reported rather recently in the context of sorption calorimetry.8−11 Very recently, the same method has been applied also to surfactants closely related to the C n TAPA p complex salts studied here, namely, C 16 TA + surfactants featuring mono-, di-, tri-, and tetracarboxylate counterions,12 and we will make comparisons with the results of the latter study where relevant. However, we have been unable to find any published study where the kinetics of water exchange between a hydrated surfactant film and an ambient humid atmosphere has been studied and analyzed. On the other hand, the instrumentation and experimental protocol that we use in the present work are quite standard for kinetic studies of the water vapor sorption/desorption of small sample quantities in general. Accordingly, they have been used in the study of similarly thin (10−100 μm thickness range) hydrated films of a range of materials, including Nafion and other (charged) polymers, starch, and gluten.13−18 Significantly thinner films (≤1 μm) can be studied using a quartz crystal microbalance (QCM) instrument, and this method has also recently been used for the precise study of water sorption/desorption kinetics for a variety of hydrated films.19−25 Notably, however, the rate of transport of water from the gas to the film surface was not measured, sometimes not even considered, in the cited studies. As far as we can judge, the characteristic features that we find for sorption/desorption kinetics governed entirely by transport through the vapor phase are not well-known among experimentalists performing similar studies on other film materials. Thus, in addition to providing new information on hydrated, responsive CS films specifically, our study should be of general relevance for the interpretation of the kinetics of water sorption/desorption for thin films of hydrated materials.

Film Casting. To enable direct comparisons with our earlier experiments, we cast films on mica sheets using exactly the same protocol as in our previous studies.1−3 Films resulting from this protocol were thoroughly characterized in ref 1. A known mass of each CS was dissolved in a known volume of ethanol to produce a solution of ca. 10% by weight. Surface films were made at room temperature by spreading an appropriate amount of solution with a pipet on a freshly plasma-cleaned mica sheet with an approximate size of 1 cm × 1.5 cm. The size of the sheet was chosen so that the mica sheet could fit horizontally on the top of a pan in the DVS instrument (see below). For CS with the short polyion the solution was added in portions of 10 μL, to ensure that the mica sheet was not flooded by the solution. The portion size for the considerably more viscous solutions of CS with the long polyion was 25 μL. For all films, the ethanol was allowed to evaporate before the next portion was added. By varying the spread amount, surface films of three different thicknesses were produced, and they are hereafter denoted as “thin”, “normal”, or “thick”. From the known deposited masses of ca. 2, 4, or 12 mg of CS, respectively, and assuming a density of 1 g/cm3 for a dry complex salt,1,5 we calculate that the approximate dry thicknesses of these films were 13, 26, and 80 μm, respectively, whereas the surface coverages ΓCS (mass of CS per unit film surface) were 13, 26, and 80 g/m2, respectively. The exact mass deposited on each individual sheet was calculated from the known concentration and total volume of the ethanolic solution used for making the film. Sorption Measurements. Sorption/desorption measurements were performed using an Aquadyne DVS from Quantachrome Instruments. The instrument contains two microbalances, each with a maximum load capacity of 5 g and a resolution of 0.1 μg, housed in a temperature-controlled chamber with a volume of ca. 200 cm3. Each balance measures the mass gain, or loss, over time. Both microbalances were used in every experimental run, with a CS-coated film placed horizontally on top of each of the balance pans. Such a parallel run typically involved two CS containing the same surfactant ion, but polyions of different length. The RH in the DVS instrument is varied by a flow of mixed wet and dry nitrogen, premixed at different ratios before entering the chamber. The gas enters through an inlet situated at the back of the chamber, behind the RH sensor. The maximum RH value that can be obtained at 25 °C is approximately 95%; higher values will result in the condensation of water in the balance chamber. Two different sorption programs were implemented in this study. Both were performed at 25 °C with a weight reading every 5 s. The “standard” sorption measurement, featuring a ramp in steps of 5% RH from 0 to 95% and back, started and finished with a drying step where the RH was brought as closely as possible to zero (normally 0.7−1% RH) during 360 min. For all other steps the maximum step duration was 180 min (except for the step 90−95% RH where the maximum duration was 360 min), with the condition that the next step was engaged if the weight change per minute was less than 0.0001% of the starting mass. The flow in the standard experiment was controlled by a standard instrument program to give transient minimum/ maximum flows of 30/200 cm3(STP)/min at the beginning of each step in order to achieve efficient changes of RH. At each RH plateau, the flow was ca. 50 cm3(STP)/min, except at the lowest levels of RH, where lower flows were used. A second type of experiment featured a faster cycling program, alternating between 45 and 95% RH during three cycles, with

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EXPERIMENTAL SECTION Materials. Polyacrylic acids (Aldrich) of MW 1800 g/mol (PAA25) or 450 000 g/mol (PAA6000) were purified by dialysis against Millipore water (5 days, Spectrum Laboratories membranes, MW cutoffs of 500 or 10000, respectively). Dodecyltrimethylammonium bromide (TCI Europe, purity > 99%) and cetyltrimethylammonium bromide (Merck, PAgrade) were used as received. All CS were prepared by titrating the chosen poly(acrylic acid) with the hydroxide form of the chosen surfactant, prepared by ion exchange, as described elsewhere.5 6779


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Figure 1. Outputs with time from a standard sorption experiment for a normal C12TAPA25 film. The complete run is shown in panel a, while panels b and c are close-ups of two steps, from 55 to 60% RH and from 90 to 95% RH, respectively. Red curves are the experimental sorption data, and green curves show the corresponding exponential fits. Black curves show the RH measured in the sorption balance.

Figure 2. Sorption isotherms recorded for all systems. Open and filled symbols refer to sorption and desorption branches, respectively. See text.

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60 min at each RH, starting and finishing with drying steps of 60 and 120 min, respectively. Here the minimum/maximum flows were again set to 30/200 cm3(STP)/min, and no cutoff for weight change per minute was used.

RESULTS Standard Sorption Experiment. Figure 1 shows an example of the output of a standard sorption experiment in the present study. In the figure, the black lines show the 6780


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plateaus in the mass, followed by rapid increases, appear in the sorption branch of the experiment. Detailed studies by sorption calorimetry have revealed a series of stable crystal hydrates for this compound, the existence of which limits the water uptake at increasing RH.12 Presumably, the more hydrated liquid crystalline states persist as metastable states on desorption, which is why the corresponding desorption values lie above those obtained in the sorption branch. For C12TAAc, on the other hand, smooth sorption isotherms (except for a small step between 5 and 10% RH), with little hysteresis are observed. Among the four studied CS, it is clear that the water sorption at a given value of RH increases both with decreasing surfactant tail length and with decreasing polyion length. This is shown in the comparison in Figure 3, which, for clarity, only includes the

measured RH and the red curves the measured water content of the film, expressed as the mass ratio between water and complex salt, mw/mCS. After the initial drying, the RH was varied in 5% increments, first by increasing RH in the sorption branch of the experiment from 0% to the maximum value of 95%. After this, the RH was stepwise decreased to 0% in the desorption branch of the experiment. Panels b and c of Figure 1 illustrate that each RH change was, to a good approximation, a step change (a stable level of RH was established rapidly) except for the final increase in the sorption branch, from 90 to 95% RH. Following each stepwise change in RH, there was a change of the water content in the film toward a new equilibrium level. For most steps, this process was excellently described by a single-exponential function, as shown in Figure 1b. For some steps, however, systematic deviations from a single-exponential dependence were noticed; an example is given in Figure 1c. Sorption Isotherms. From the final plateau values of mw/ mCS, recorded after each step change in RH and immediately before the next (see Figure 1), sorption isotherms were constructed. These are shown in the graphs in Figure 2, one for each investigated system. Each graph includes sorption and desorption data for thin, normal, and thick films. The recorded sorption isotherms show a satisfactory overall reproducibility and independence of the thickness of the studied film. The variations that do occur for each system show no systematic correlation with the film thickness. On the other hand, small systematic differences are seen between the sorption and the desorption isotherms, where the latter systematically ends up at a lower final dry mass. We attribute this difference to a more complete drying of the CS at the end of the desorption branch, compared to the value obtained in the more rapid initial drying step before the start of the sorption branch of the cycle. Therefore, the mass at the end of the desorption branch is here used as the reference dry mass mCS in the calculations of mw/mCS. We must cautiously allow for the possibility that there could be residual water left even in this extensively dried material. However, if that should be the case, the proportion of residual water is independent of the film thickness in the investigated range, since the sorption isotherms do not vary systematically with film thickness. For all CS, the obtained isotherms are smooth and featureless. A likely reason is that, over most of the studied RH range, the CS exist in one or the other of two closely related liquid crystalline structures: the 2D-hexagonal structure, featuring extended cylindrical surfactant aggregates, at high degrees of hydration, or the rectangular or “distorted hexagonal” structure, featuring flattened rodlike aggregates with a noncircular cross-section, at low degrees of hydration.1,2 At coexistence, the rectangular and hexagonal phases are not expected to differ much in water content;26,27 this explains the absence of a visible step at the phase transition in the sorption isotherm. At very high water activities, a transition to the Pm3n micellar cubic phase occurs for all CS,1 but this transition is not expected to be reached in the current experiments, except possibly for the C12 complex salts (see below). For the C16 complex salts, previous osmotic decompression experiments on films of the same type as those investigated here have shown that the transition 2D-hexagonal → micellar cubic occurs at water activities corresponding to >97% RH.3 For the acetate surfactants the picture is less simple (Figure 2). For C16TAAc, sorption isotherms of two different films show a reproducible hysteresis, where at least three distinct

Figure 3. Water sorption isotherms for all in investigated CS, obtained on desorption of thick films. Symbols refer to C12TAPA25 (unfilled blue circles), C16TAPA25 (filled blue circles), C12TAPA6000 (unfilled red squares), and C16TAPA6000 (filled red squares). Data at 100% RH, taken from ref 7, show concentrations of the maximally swollen cubic phases of C12TAPA25 and C12TAPA6000 in equilibrium with pure liquid water. Error bars represent the experimental resolution in the determinations of the phase boundaries.

desorption isotherms for thick films: the water sorption of the CS increases in the order C16TAPA6000 < C16TAPA25 ≈ C12TAPA6000 < C12TAPA25. This pattern is seen at all studied RH and was also seen for thin and normal films. In Figure 3, we have included previously obtained data for C12TAPA6000 and C12TAPA25 at RH = 100%. The latter data are taken from a detailed study of phase equilibria of C12 complex salts7 and show the compositions of fully swollen bulk phases of CS in equilibrium with excess pure water. Clearly, the latter data points follow smooth continuations of the respective sorption isotherms, as predicted by fundamental thermodynamics. Kinetics of Water Exchange. Panels a and b of Figure 4 give an overview of the time constants, τ, obtained from the exponential fits (see Figure 1) to the time-dependent mass changes following all 5% step changes in RH for the two “extreme” CS of our study, that is, C12TAPA25 (short surfactant tail and short polyion) and C16TAPA6000 (long surfactant tail and long polyion). For each transition step, data are plotted versus the higher of the two RH values between which the step change was made. For reasons that will become apparent 6781


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sets run in parallel in the logarithmic plot) over an intermediate range of RH, but the extent of this range differs among the CS. We will return to the significance of this observation in the Discussion. Below the range of proportionality, τ generally increases more rapidly than the relative mass change with decreasing RH. Sorption/Desorption of Films in Cycles of Large RH Changes. To test the robustness and reproducibility of the water sorption/desorption of our films, we performed experiments with repeated cycles of sorption/desorption of the CS and surfactant films in large steps, between 45% and 95% RH. Figure 5 shows an example for a normal C12TAPA25 film. The

Figure 4. Graphs for all investigated films of C12TAPA25 (a) and C16TAPA6000 (b), showing fitted time constants for all sorption/ desorption steps in red and the corresponding changes in hydration at the end of each step in black. For each step, the data are plotted versus the higher of the RH values between which the step change was made; see text. The different symbols denote thin (circles), normal (diamonds), and thick (triangles) films; open symbols represent sorption data and filled desorption data.

below, we also include in the same graphs the plateau changes in water content, Δmw/mCS, that correspond to each step change in RH. The τ and Δmw/mCS data are plotted using two logarithmic y-axis scales with the same total expansion (a factor of 200), to facilitate comparison. For each system, data for sorption and desorption are shown for thin, normal, and thick films. These data illustrate all the features and trends that we have observed in this study. The corresponding plots for the “intermediate” polyion−surfactant ion combinations (C16TAPA25 and C12TAPA6000) and the acetate surfactants are provided as Supporting Information (Figure S1). The relative mass changes Δmw/mCS are (as expected from Figure 2) essentially independent of film thickness and only very weakly dependent on the direction of change (sorption or desorption). Overall they vary smoothly with RH, with a shallow minimum in the low-RH region. For C12TAPA25, a reproducible steep increase in Δmw/mCS appears from 85 to 90% RH, which likely indicates a transition to the cubic Pm3n phase.7 A similar steep increase appears also for the final 90− 95% RH step in C12TAPA6000 (see Figure S1 in the Supporting Information), but is absent for the C16 CS. Like the relative mass changes, the sorption/desorption time constants vary with RH with a minimum at intermediate RH values. In addition, they show a strong dependence on the film thickness: τ increases with increasing film thickness. There is consistent hysteresis in the data so that the time constants for sorption are longer at high RH, whereas the time constants for desorption are longer at low RH. The hysteresis is small in an intermediate range of RH, but increases strongly on approaching the extremes of the RH range. Notably, there is a striking proportionality between τ and Δmw/mCS (the data

Figure 5. (a) Repeated large step water sorption/desorption cycles between 45 and 95% RH for a normal film of C12TAPA25. (b) Expanded plot of one cycle, comparing experimental (red) and fitted single-exponential (green) curves.

large changes in hydration represent switches between one environment with a water activity similar to ambient air and another with the highest water activity that could be reached with the experimental equipment. The desorption step of such a process mimics rapid dehydration experiments that we recently performed on similar CS films, while continuously recording synchrotron SAXS profiles from the films.2 The accompanying changes in hydration recorded after the step changes of RH in the sorption balance were, as above, fitted to single-exponential expressions. Systematic deviations between the experimental curves and the fitted exponentials were this time seen, but the reproducibility of the fitted τ values (provided as Table S1 in the Supporting Information) was quite good, within a few percent. Figure 5 and Table S1 refer to normal films; experiments were also performed on thin and thick films with the same trends being observed. In each experiment, the characteristic time for the sorption process was much longer than for the desorption process. The time constants for equilibration of the water content on dehydration (4−5 min) agree well with our previous observation that the changes in the internal liquid crystalline structure of a film, 6782


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his seminal theoretical analysis of the phase behavior of CnTAPAp complex salts mixed with water, Hansson used the Flory−Huggins (FH) theory to capture the effect of polyion chain length on the water uptake.28 The FH theory captures, on a mean-field level, the increase in the entropy of mixing for a polymer solution when the degree of polymerization decreases at a fixed volume concentration of polymer. Hansson found that the theoretical dependence of the phase boundaries on p was small beyond p = 10, which is at variance with our results. Thus, the simple FH model seems to underestimate the effect on the osmotic pressure of CS of increasing the polyion length, also for p > 10. The FH theory only considers the change in the ideal contribution to the entropy of mixing of the chains (the change in the number of molecules), but for polyions “sandwiched” between surfactant aggregates there should also be a contribution to the osmotic pressure from the additional configurational degrees of freedom of the free chain ends, which are much more frequent at p = 25 than at p = 6000. In addition, we cannot rule out a possible contribution from chemical differences of the end residues, originating from the chain initiator in the polymerization reaction.4 Also the (small) remaining influence of the surfactant tail length in Figure 6 appears to be a real effect. We recall that over the studied range of RH, the hexagonal and rectangular phases dominate, so we should seek the explanation in the properties of long surfactant rods. The stronger curvature of the C12 rod leads to a larger average distance between the charged head groups on the rod surface. Moreover, thermal fluctuations on all length scales, from a protrusion of the individual surfactant ions to “undulations” of the rod, should be stronger with a decreasing length of the surfactant tail.29 Both these effects would lead to a larger water uptake for the C12 complex salts. Kinetic Model for Water Exchange Limited by Diffusion in the Gas Phase. A major part of our experimental results are detailed data on the sorption/desorption kinetics for varying film thicknesses over the studied RH range. We will devote the remainder of the Discussion to interpreting these data and discussing the wider implications of our interpretation. Our first objective is to understand the striking features that we typically (but not universally) observe (Figures 1b, 4, and S1) for the time-dependent sorption/desorption of water following a small, stepwise change in RH. These are (1) a singleexponential variation of the water content that is (2) independent of the direction of change (sorption or desorption) but (3) proportional to the slope of the binding isotherm over the investigated step change in RH. For this purpose we will here digress to a detailed derivation of an expression for the time-dependent water exchange between a surrounding reservoir of humid gas and a hydrated film of thickness d, after an instantaneous change of the water activity (the relative humidity) arw in the gas reservoir; see Figure 7. Anticipating the results of our analysis, we will assume the following limiting conditions. (1) The time-dependent net transport of water between the film and the gas reservoir can be modeled as a one-dimensional diffusion through a stagnant gas layer of thickness δ next to the film. No variation in δ along the film or with time is considered, and the relative humidity of the reservoir is assumed to be kept constant, through convection, after each step change. (2) The film is sufficiently thin so that the equilibration of the water content within the film along the film normal is rapid; no gradient in water concentration develops in the film as a result of water exchange with the gas reservoir.

following a large stepwise decrease in the RH of the environment, leveled off after 10−20 min.2

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DISCUSSION Water Sorption Capacity of Complex Salts. The water sorption isotherms show that, for all values of the relative humidity, the equilibrium water uptake of the various CS increases in the order C 16 TAPA 6000 < C 16 TAPA 25 ≈ C12TAPA6000 < C12TAPA25 (see Figure 3). We will now discuss the possible origins of this sequence. The ionic groups in the system should give the dominating contributions to the water uptake, since the CS contain no other hydrophilic groups. In Figure 6 we therefore present the data from Figure 3

Figure 6. Hydration data from Figure 3 expressed as the number of water molecules per ion pair of the complex salt. Symbols refer to C12TAPA25 (unfilled blue circles), C16TAPA25 (filled blue circles), C12TAPA6000 (unfilled red squares), and C16TAPA6000 (filled red squares).

recalculated as the number of water molecules sorbed per CS ion pair, using the theoretical molar masses of water and of the repeat units of C12TAPAp and C16TAPAp. We now find that the relative differences between the CS have decreased significantly, compared to those shown in Figure 3. Interestingly, we also note that the water sorption per ion pair of CS is strikingly similar to that previously found for thin films of Nafion and sulfonated poly(ether ether ketone),16,19,22 supporting the notion that the essential source of the water uptake in all these different systems are, in fact, the ion pairs. There remain, however, some systematic differences among the CS in Figure 6, with the sorption per ion pair increasing in the order C16TAPA6000 < C12TAPA6000 < C16TAPA25 < C12TAPA25. The same sequence was seen also for thin and normal films (not shown). We will begin by discussing the possible effect of the polyion chain length, which dominates the remaining quantitative differences in Figure 6. Previous SAXS studies of CnTAPAp in water have shown that the dimensions of the surfactant aggregates in liquid crystalline phases depend essentially only on the water content, and not on the choice of counterion (long or short polyacrylate, or acetate).5−7 Thus, we conclude that the larger water uptake of the CS with short counter-polyions is a direct effect of the polyion chain length. In 6783


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Figure 7. Schematic picture of the model for water exchange between a hydrated film and a surrounding humid gas reservoir, following an instantaneous change of the relative humidity in the reservoir. The water activity is assumed to be uniform within the film and in the humid gas reservoir, respectively, and to vary only in a stagnant layer of the gas between the gas reservoir and the film.

(3) Equilibration of water across the film−gas interface is also rapid so that the (uniform) water activity in the film interior is always the same as in the interfacial layer of gas in immediate contact with the film surface, where it is given by asw. (4) The humid gas is an ideal gas mixture so that the concentration of water, cgas w (in moles per unit volume), is given by cwgas =

pw RT

=

where ΓCS is the surface coverage (mass per unit area) of complex salt on the substrate. Mass balance across the film−gas interface (combining eqs 2 and 3), and conversion of molar concentration to mass concentration (with Mw denoting the molar mass of water), gives M w Dwgaspw0 r d mw = (aw − aws) dt mCS ΓCSδRT

awpw0 (1)

RT

To reduce the number of unknowns, we must relate the relative humidity of the gas to the hydration of the film. If equilibrium is always maintained across the film−gas interface, and if the equilibration of the water content within the film along the film normal is rapid, as assumed above, the relation between asw and mw/mCS at any point in time is given by the sorption isotherm. Moreover, for sufficiently small changes in RH, this isotherm is effectively linear (see Figure 2). Under these conditions we obtain

p0w

Here pw is the partial pressure and the saturation pressure of water at the relevant temperature T, aw (≡pw/p0w) is the water activity, and R is the ideal gas constant. Fick’s first law, valid for an ideal gas, then gives for the flux Jw,x of water (in moles of water per unit area) in the x direction, normal to the film, Jw , x = −Dwgas =−

Dwgaspw0 δRT

∂cwgas c gas(d + δ) − cwgas(d) = −Dwgas w ∂x δ (awr

−

aws)

⎛m m (aws(t ) − awr) = K −1⎜ w (t ) − w mCS ⎝ mCS

(2)

where we have used eq 1 in the final equality and chosen the positive x direction in the direction away from the film; that is, the flux is positive for evaporation. Dgas w is the diffusion coefficient of water in the gas at the relevant temperature and pressure. Turning to the situation inside the film, we will express the water concentration as the mass ratio between water and complex salt in the film, mw/mCS, as in the experimental sorption isotherms. The total rate of evaporation, in mass of water per unit time, from a film with a total area A is then d d mw d mw mw = mCS = A ΓCS dt dt mCS dt mCS

(4)

where asw(t)

mw (t ) mCS

⎞ ⎟ ⎠ eq

(5)

is the equilibrium water content of the film at

and where, at sufficiently long times after the step change,

asw(t) → arw and

mw m (t )→ m w mCS CS

. In eq 5 we have introduced the ∞

(constant) slope of the sorption isotherm in the relevant interval of RH, K=

Δ(mw /mCS ) Δaw

(6)

We can then rewrite eq 4 to obtain the simple differential equation

(3) 6784


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The Journal of Physical Chemistry B M w Dwgaspw0 −1⎛ mw m d mw =− K ⎜ (t ) − w dt mCS ΓCSδRT mCS ⎝ mCS

⎞ ⎟ ⎠ ∞

To couple the external mass transfer in the gas phase to the internal mass transfer in the film, one must, as in our derivation above, relate the concentration difference across the stagnant gas layer to the corresponding concentration difference in the film at the interface, via the slope of the sorption isotherm. This results in a new mass transfer coefficient kc, where the surface concentration of water in the film, rather than the interfacial concentration in the external gas phase, is used as a driving potential. A detailed description is given, for instance, in a recent paper by Saeidpour and Wadsö.30 We note also that by combining eqs 9 and 10 in the latter work, one obtains an equation that is equivalent to our eq 7. In principle, therefore, the analysis of the water sorption/desorption process that we present here is not new. That being said, we have not found in any previous work the explicit expressions 7, 8, and 9 that follow from the analysis. We are here specifically considering a case where the external mass transfer through the stagnant gas layer is rate-limiting. In the engineering literature, the ratio between the internal and the external mass transfer resistance is commonly described by the mass transfer Biot number, Bim, often expressed as

(7)

With the appropriate boundary condition at t = 0, the familiar solution is an exponential variation of the film water content with time, ⎛m mw m (t ) = ⎜ w (0) − w mCS mCS ⎝ mCS

⎞ m ⎟ exp( −t /τ ) + w mCS ⎠ ∞

∞

(8)

with a characteristic time constant τ given by τ=

ΓCSδRT M w Dwgaspw0

K (9)

We note that eqs 8 and 9 predict exactly the features 1−3, described at the beginning of this section, which we observed over more or less wide RH ranges in our kinetic experiments. All parameters appearing in the expression for τ are experimentally accessible in principle, the most difficult one being the stagnant-layer thickness, which depends on the detailed, often unknown, gas flow around the film. The only material-dependent parameter entering τ is an equilibrium property, the slope of the sorption isotherm. This is a consequence of the assumptions of rapid equilibration inside the film and across the film−gas interface; as long as these assumptions are valid, we need not know any details about the rate of water transport in the film. Given, as we assume, that the transport of water from the reservoir to the surface of the film is the rate-determining step for changes in hydration, we obtain the strong prediction that for experiments where both ΓCS and δ are kept constant, the time constant for water exchange should only vary with K. This should hold for the same film at different positions of the isotherm, or for films of different compounds, for instance, surfactant or CS. The stagnant layer can be likened to a “pump” that pumps water at a certain instantaneous rate, which is completely determined by the instantaneous conditions in the gas outside the film (the difference arw − asw), the temperature, and the thickness of the stagnant layer. The change in asw with the amount of water pumped is, however, coupled to the film via the sorption isotherm. If the latter is steep, much water needs to be pumped into the film in order to change the activity of water in the film and, consequently, change asw. Therefore, the approach to equilibrium becomes slow. If the film thickness (the surface coverage ΓCS) increases, the same reasoning applies, that is, more water needs to be transported to or from the film to change asw. Before concluding the present section, we will for later use make contact with descriptions of the above considered sorption/desorption situation in the engineering literature. The molecular diffusion of water through a gradient in an external surface gas layer is there commonly described as an external mass transfer resistance, characterized by a mass transfer coefficient kv (m s−1), where the subscript v (for vapor) signifies that the gradient is described in terms of the concentration of water in the gas. In our stagnant-layer description, the mass transfer coefficient is simply

kv =

Dwgas δ

Bim =

kLc Dw

(11)

where k is the appropriate external mass transfer coefficient, Lc is a characteristic length in the hydrated material (here naturally taken as the film thickness d), and Dw is the water diffusion coefficient in the material. The case where the internal equilibration of the water content is rapid compared to the mass transport of water between the gas reservoir and the film corresponds to the limit Bim ≪ 1. It is possible to show that, for the situation considered here (rapid equilibration across the gas/film interface), and using our notations, Bim =

d2 τDwfilm

(12)

where is the diffusion coefficient of water in the film. d Physically, the ratio d2/Dfilm w ≡ tw is the time that it takes for a water molecule to diffuse a root mean square (rms) distance equal to the film thickness. Thus, we can identify, for the situation desribed in Figure 7, the Biot number as the ratio between the characteristic diffusion time tdw across the film and the time constant τ, defined through eqs 8 and 9. Rapid internal equilibration by diffusion, as assumed in our derivation, corresponds to tdw ≪ τ. Finally, we will use eqs 9 and 12 to derive a useful scaling relation for the Biot number. In the limit considered, we found that τ is proportional to the thickness of the stagnant layer δ and, also, to the surface coverage ΓCS. The surface coverage is, in turn, proportional to the film thickness d. Inserting τ ∝ δd in eq 12, we obtain, at constant temperature, Dfilm w

Bim ∝

d δ

(13)

Comparisons between Experimental Results and Model Predictions. The analysis resulting in eqs 8 and 9 leads to a number of useful predictions that we will now summarize and compare with our experimental results. (1) For a film initially in equilibrium with a reservoir of humid gas, the water content of the film will exponentially approach a new equilibrium value after a small instantaneous step change ΔRH of the RH in the reservoir, provided that the

(10) 6785


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The Journal of Physical Chemistry B water sorption isotherm for the film is linear over the relevant ΔRH. (2) For a given ΔRH, the same value of τ should be obtained regardless of the direction of the change (sorption or desorption). (3) The time constant τ should be proportional to the slope K of the sorption isotherm over ΔRH. (4) For a given surface coverage, other experimental conditions being the same, the constant of proportionality between τ and K should be system-independent. (5) The time constant τ should be proportional to the surface coverage ΓCS (or dry thickness) of the film. To check more closely the validity of predictions 2−4, we show in Figure 8 the ratio τ/K, calculated from our

experimental data, across the RH range for selected films. In each of the two graphs a and b of Figure 8, we compare systems with the same nominal surface coverage. In Figure 8a, we have chosen the data for thick films of all CS. The data for thick films show the least scatter and, moreover, reveal the differences between the various CS most clearly. (Ratios for all thicknesses of all CS are presented as Figure S2 in the Supporting Information.) For C12TAPA25 and C16TAPA25, the ratios indeed converge to a common, constant value with an average of 15.0 min over a wide range of RH, roughly 40−85%. This is not a trivial result, since the individual data entering the ratio show a substantial variation: over this region, the slope K varies by a factor of 2.7 for C12TAPA25 and differs by a factor ≥ 1.4 between C12TAPA25 and C16TAPA25. Over the same RH range, a plateau is observed also for C12TAPA6000, but at a higher mean value (19.4 min). Finally, over a much smaller range (65−85% RH), the τ/K ratio for C16TAPA6000 plateaus around a mean value (20.1 min) very similar to the plateau value for C12TAPA6000. In summary, we indeed observe constant τ/K ratios in an intermediate RH range of our films, but there is a positive deviation at low RH that sets in at a higher RH for an increased length of the polyion, or an increased length of the surfactant alkyl chain. However, depending on the polyion length, there seem to be two different plateau values for τ/K, which is not predicted by our theory. A factor that at least contributes to the larger plateau value for the PA6000 CS is that the ethanol solutions of these CS were quite viscous and, hence, we did not manage to spread the solutions over the entire surface of the mica sheet, all the way up to the edges. As a consequence, the film thicknesses of the PA6000 CS were, in fact, significantly and systematically larger than for the low-viscous PA25 CS. We therefore decided to include also a simple acetate surfactant in our comparison, and we chose C12TAAc, which shows the least complex sorption isotherm for the surfactants (see Figure 2). Figure 8b compares the results for normal films of C12TAPA25 and C12TAAc, and although the noise is slightly larger than for thick films, it is quite clear that the τ/K data converge on a single plateau at intermediate RH for the two systems. Data for all normal CS films were also compared in a plot similar to Figure 8a (not shown), showing the same trends as in Figure 8a, but with more scatter in the data and a longer extension of the plateau for C16TAPA6000 toward low RH values. Turning to the effect of film thickness, prediction 5 agrees with the trends seen in Figure 4, but the agreement is not quantitative. In Figure 9, we show data of τ/mCS for the three films of C12TAPA25 (the surface areas of the substrate mica sheets were, within error, the same). Clearly the normalized time constants decrease systematically with increasing film thickness. Systematic deviations of varying magnitudes, but following the same trend as in Figure 9, were obtained for the three other CS (see Figure S3 in the Supporting Information). We will return to this observation below. Surface Mass Transfer Coefficient and the Stagnant Layer. The plateaus in Figure 8 identify conditions, in our DVS experiments, where data behave in accordance with the model predictions. We can use these data for an explicit calculation of the stagnant-layer thickness. Specifically, we will use the average value of 15.0 min for the ratio τ/K obtained from the extended plateaus for C12TAPA25 and C16TAPA25 in Figure 8a (this choice will be justified below). Inserting this value in eq 9, together with known constants and the values ΓCS = 7.7 × 10−2 air −5 kg/m2 (thick film), Dgas m2/s (from w = Dw (25°C) = 2.6 × 10

Figure 8. Comparisons of the ratios τ/K as functions of RH on sorption (unfilled symbols) and desorption (filled symbols) for various investigated films. (a) Thick films of C12TAPA25 (blue circles), C16TAPA25 (green triangles), C12TAPA6000 (red squares), and C16TAPA6000 (brown diamonds); (b) normal films of C12TAPA25 (blue circles) and C12TAAc (purple squares). 6786


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At this point we should note that the existence of a barrier, or “skin”, at the film−gas interface would contribute to the resistance to mass transfer between the gas reservoir and the film. If such a layer exists, preventing a rapid equilibration across the interface, it would affect the measured overall experimental mass transfer coefficient and, thus, our 22,35 interpretation kv = Dair Indeed, w /δ above would be incorrect. we suspect that the latter situation might arise under some circumstances for very low RH; see below. However, there are compelling arguments against the existence of a significant interfacial barrier at higher RH, under the conditions where we evaluated δ, kv, and kp above: in Figure 8, we have demonstrated that the ratio τ/K is (i) constant over a large RH range and (ii) system-independent, at least for the three systems C16TAPA25, C12TAPA25 and C12TAAc cast from low-viscous solutions. We regard it as unlikely that the latter three systems, among which both the surfactant tail length and the nature of the cation (monomer or polymer) vary greatly, should form interfacial barriers that give rise to identical mass transfer resistances over a wide range of RH. Turning to the literature, we have not found any study on the possible existence of barriers for water transport at the interface between concentrated surfactant (or CS) systems and a vapor phase. For dilute systems, a study by Kuznetsov et al. showed that the effect on water evaporation of films of polyelectrolyte−surfactant complexes spread on the water surface was marginal; the reduction was 6 and 8% for complexes of polystyrenesulfonate/dodecylammonium and poly(diallyldimethylammonium)/docecyl sulfate, respectively.36 When and Why the Model Does (Not) Predict the Experimental Results. If the assumptions behind our limiting model were correct throughout the investigated RH range, then all data in each of panels a and b of Figure 8 would have collapsed on a single horizontal line. Similarly, all data in Figure 9 would have collapsed on a single curve. Clearly, this is not the case, and here we will conclude that some of the observed deviations reflect real properties of the films, whereas others probably reflect limitations of our experimental conditions. We will begin with the former, since these contain significant information on the various CS films and the differences between them. Deviations at Low RH. Toward lower RH, starting somewhere in the interval 25−60% RH (depending on the system), the τ values start to increase more rapidly with decreasing RH than predicted by K. Most probably, this is because a basic assumption of our model does not hold for very dry films: tdw is no longer negligible compared to τ. To estimate tdw, we will use the limited data available on the diffusion of water in very concentrated CS systems. Svensson et al. reported a self-diffusion coefficient of 4 × 10−10 m2/s of water, obtained from NMR pulsed field gradient experiments, in the cubic phase of C16TAPA30 containing 55 wt % water.37 In an unpublished study using the same technique, Söderman at our laboratory (personal communication) obtained a water selfdiffusion coefficient of 1 × 10−10 m2/s in a sample of C16TAPA30 containing 30 wt % water. Using the latter value and a thickness of ca. 100 μm for a hydrated thick film, we obtain the characteristic time tdw = 50 s for a water molecule to diffuse an rms distance equal to the film thickness. For a thick film of C16TAPA25 we here found a water content of ca. 30 wt % at 85% RH (Figure 3), and for the corresponding desorption step at 85% RH, the measured time constant τ was ca. 800 s. Thus, there is more than an order-of-magnitude difference between the two characteristic times (Bim = 0.06), confirming,

Figure 9. Mass-normalized water sorption time constants τ/mCS (see text) for thin (black), normal (red), and thick (green) surface films of C12TAPA25. Open symbols indicate sorption data; closed, desorption data.

ref 31) and p0w(25°C) = 3.167 × 103 Pa (from ref 32), gives a stagnant-layer thickness δ = 7.0 mm. (We note in passing that this thickness is 1−2 orders of magnitude larger than the film thickness. Hence, any roughness of the dry film surface1 will be insignificant compared to the stagnant layer and will have no consequence for the kinetics of water exchange.) Alternatively, we can choose to calculate (see eq 10) the mass transfer coefficient kv = 3.7 × 10−3 m/s. Yet another mass transfer coefficient commonly used in the literature is kp (kg/m2sPa), where the partial pressure of water in the gas, rather than the water concentration (cair w in our notation), is used to describe the water content. The relation between kp and kv is obtained by using the ideal gas law: kp = kvM w /RT

(14)

Using eq 14 together with eqs 10 and 9, we obtain kp = 2.7 × 10−8 kg/m2s Pa from the chosen data under our experimental conditions. It surprises us, especially in view of our results, that attempts to actually determine the external mass transfer resistance provided by the stagnant gas layer (the parameter δ, kv, or kp) in the setups used in previous experimental studies appear to be scarce. However, a careful analysis of the external mass transfer resistance in a DVS instrument was recently performed by Wadsö et al., who used saturated salt solutions at 20 or 25 °C as sorption samples for the purpose of validating the RH generation in the instrument.33 The authors point out that the flow pattern in the instrument is complex and not necessarily uniform across the sample surface; hence, they preferred to determine the product kpA, which may be regarded as an overall mass transfer coefficient. They obtained kpA = 1.7 × 10−12 kg/(s Pa) for a sample with a surface area A ≈ 5 × 10−5 m2. From these values, one obtains kp = 3.4 × 10−8 kg/(m2 s Pa) in their experiments. The latter value is very similar to the value 2.7 × 10−8 kg/(m2 s Pa) that we calculated above from our experiments, lending credibility to our analysis. Wadsö has earlier published a review collecting calculated and experimentally determined kp values for a large number of systems under a range of external gas flow velocities.34 A kp of 3 × 10−8 kg/(m2 s Pa) is at the lower end of these values, corresponding to gas velocities of the order of 0.1 m/s or less in the case of laminar flow. Again, this seems reasonable, since the gas flow was quite gentle in our experiments, with a gas flow of ca. 50 cm3/min through a DVS chamber with a volume of ca. 200 cm3. 6787


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be significantly different in the desorption and sorption experiments, likely influencing δ; (iii) the recorded time dependence of RH in the sorption experiment deviated significantly from a step function (see RH data for the change 90 → 95% in Figure 1c). The same three factors, all of them amplified, should contribute to the differences between the sorption and desorption time constants obtained in the experiments involving large step cycles in RH (Figure 5 and Table S1 in the Supporting Information). Only factor (i) reflects a property of the film; the other two result from significant differences at high RH between the experimental conditions and the model assumptions. Depencence on Film Thickness. The final deviation between our model and the experimental data concerns the predicted proportionality between the time constant and the surface coverage ΓCS (or dry thickness) of the film. We have no conclusive explanation for this deviation, but we do have indications that variations in the gas flow in the sorption balance contributes significantly. The total flow is normally optimized to produce, as closely as possible, a step change of RH, followed by a flat plateau. In order to achieve this, there are large transient variations in the total flow at the beginning of each step: a transient peak in the sorption branch, and a transient dip in the desorption branch, each followed by a plateau of constant flow. The initial transients could give rise to some hysteresis between sorption and desorption. However, for thick films, with larger time constants, the initial transients should be a smaller perturbation, due to the longer duration of the plateau flow. For this reason, and in view of the smaller scatter in the experimental data, we would regard the data for the thick films as the most reliable in our experiments. Therefore, we used such data in the calculations of the parameters δ, kv or kp above. We made some unsuccessful attempts to measure under strictly constant flow conditions during the entire isotherm, by locking the settings of the equipment to a constant, high flow. However, this setting of the equipment resulted in large fluctuations in RH and a large scatter in the data of τ and Δmw/mCS versus RH. Wider Relevance of Our Analysis. Our results show that water transport through a stagnant gas layer can completely determine the kinetics of water sorption/desorption processes for sufficiently thin and sufficiently hydrated films measured under conditions of gentle gas flow, as in a DVS instrument. What “sufficiently thin” and “sufficiently hydrated” actually mean will clearly depend on both the gas flow conditions and the material of the film; see the expressions for the Biot number (eq 12) and the time constant τ (eq 9). However, our finding, that Bim ≪ 1 at hydration levels down to ca. 3 water molecules per ion pair for ca. 100 μm thick films of CS in our DVS instrument can serve as rough guidance. The scaling Bim ∝ d/δ, eq 13, shows, furthermore, that for the very thin films that can be studied by QCM, this limiting situation should apply for even drier films under similar gas flow conditions.22 Note also that it may be difficult to decrease the stagnant layer proportionally to the film thickness, if the latter is decreased by more than an order of magnitude. Data reviewed in ref 34 show that for laminar flow it takes a gas velocity of 10 m/s to produce a stagnant layer as thin as 0.5 mm and that δ decreases very slowly for still higher velocities. We conclude that one should be generally very cautious in interpreting kinetic results obtained from measurements such as those considered here in terms of the internal water transport in the film. This fact has recently been discussed

for the highly hydrated samples, our assumption of rapid equilibration of the water content in the film. On the other hand, given the quite rapid decrease of the water self-diffusion from 55 to 30 wt % water, it seems reasonable that Dfwilm could decrease by an order of magnitude or more toward the drier films of our study, leading to a Biot number ≥ 1. A striking feature of Figure 8a is that the upturn of the τ/K ratio on decreasing the RH commences at a higher RH for the longer polyion and the longer surfactant tail (see Figure 8a). The sorption isotherms (Figures 3 and 6) show that the water contents of the CS vary with the same factors. We therefore checked whether the slowing generally appears at some critical low hydration, independent of the CS. Scrutinizing the data in Figure 8a, we found that the plateau of τ/K ends at (approximately) 65, 35, 40, and 35% RH for C16TAPA6000, C12TAPA6000, C16TAPA25, and C12TAPA25, respectively. The corresponding hydration values nw/nCS (Figure 6) of the various films recorded at these humidities were 3.7, 2.3, 3.1, and 3.6, respectively. These values are connected with larger error bars, primarily because of the difficulty to pinpoint the end of the plateau, but nevertheless suggest that the water content is indeed a key factor governing where the condition tdw ≪ τ is no longer valid: below ca. 3 water molecules per ion pair for the thick films under the conditions of our measurements. We emphasize that one should not attribute any particular significance to this specific hydration number, since τ depends on both the gas flow conditions (δ) and the film thickness. This implies that for each CS, the upturn at low RH should commence at a lower RH, that is, a lower hydration number, with decreasing film thickness. Indeed, this trend was observed, and most clearly for C16TAPA6000 (see Figure 4b and, also, plots of τ /K for all thicknesses of all CS in Figure S2 of the Supporting Information). A consistent observation for all films (see Figures 4 and 8 and Figures S1 and S2 of the Supporting Information) is an increasing hysteresis in the time constants toward low RH: as the film gets drier, the time constants on desorption become increasingly larger than those for sorption. Tentatively, we ascribe the hysteresis to the develoment of a “skin” at the surface of the film on drying. The onset of the hysteresis is system-dependent (Figure 8), but it generally sets in at an RH somewhere in the upturn region of τ /K. Our data suggest also other differences between the CS in this upturn region. For instance, τ seems to increase more rapidly with decreasing RH for C16TAPA25 than for the other CS (see Figure 8a and Figure S1 of the Supporting Information). More detailed measurements would be required to confirm these observations. Deviations at High RH. At high RH, our experiments also show systematic deviations between the model predictions and the experiments. A significant and increasing hysteresis in the time constants sets in at around 70% RH, but now with the longer times appearing for the sorption branch of the experiment (see Figures 4 and 8 and Figures S1 and S2 in the Supporting Information). Moreover, for the highest RH step (90−95%), there is a consistent increase in the τ/K ratio especially on sorption (Figure 8). Finally, the experimental sorption/desorption data start to show systematic deviations from a single-exponential function (Figure 1). A number of factors should contribute to these observed deviations at high RH: (i) the assumption of a linear sorption isotherm becomes less accurate (Figure 2), leading to systematic deviations between the experimental data and the exponential fit; (ii) the flows generated in the sorption balance were here observed to 6788


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the film. It is demonstrated that a detailed examination of the ratio τ/K as a function of RH can pinpoint the conditions where the model assumptions do or do not apply, providing valuable information on both the conditions in the experimental setup and the properties of the investigated film. Over wide ranges of RH, the assumptions of our kinetic model were indeed found to apply for the CS films studied and the DVS instrument used. Specifically, for complex salt films of ca. 100 μm thickness, the water transport across the interfacial gas layer was found to be rate-limiting at water contents above ca. 3 water molecules per ion pair (ca. 15 wt % water). For all but the least hygroscopic CS (C16TAPA6000), this corresponds to RH values higher than 35−40%. For thinner films, the limit of rapid internal equilibration was found to apply for even less hydrated films. This has implications for the possible application of complex salt films as humidity-responsive materials: it should be eminently possible to make films where the response time at a realistic humidity of 30% or higher and at realistic gas velocities is limited by the external water transport in the gas phase and not by the internal transport of water in the film. An important reason for the efficient internal water transport is, most likely, the water-continuous nature of the liquid crystalline structures formed by the hydrated complex salts. As the RH decreases further, the assumption of rapid internal equilibration eventually breaks down, and the rate constant τ can eventually reach very high values (∼1 h) depending on RH, the identity of the CS, and the thickness of the film. To date, data on the internal structures of very dry CS are scarce, but the continuity of the water domain should necessarily disappear at some low, but finite, level of hydration. The fact, established here, that water transport in the gas phase may be rate-determining for water vapor sorption/ desorption of hydrated films of thicknesses up to 100 μm does not seem to be generally appreciated. Thus, our study has important implications for the analysis and interpretation of similar experimental studies of thin hydrated films in general.

extensively by Hansen, who showed that an apparent nonFickian water diffusion in thin films can result from erroneous analyses of the desorption/sorption process where a significant “surface condition”, that is, a resistance to the transport of solvent molecules between the gas and the surface layer of the bulk of the film, is neglected.38 Studies by other laboratories of water desorption/sorption of thin films have indeed concluded that the influence of the interfacial mass transfer resistance, originating from a stagnant gas layer and/or a barrier at the film surface, is significant or even dominating.22,39 Nevertheless, in several other recent studies, the possible influence of such a surface condition seems not to have been considered.13−16,18−21,23−25 The (tacit) assumption of a large mass transfer Biot ratio may or may not have be correct in each individual case, but our results strongly suggest that this must be checked. For such a check, the experimental design and data analysis used here are very useful, namely, to (i) analyze the kinetics of water sorption/desorption following small step changes of ΔRH with exponential fits and (ii) check across the isotherm whether the associated time constant for the water exchange is proportional to the slope of the sorption isotherm and, also, (iii) to check if it is proportional to the film thickness. As shown here, our experimental approach makes it possible to pinpoint the conditions where the assumption of rate-limiting interfacial transport does and does not apply. If, indeed, the limit Bim ≪ 1 applies, the stagnant-layer thickness of the experimental setup can actually be measured. We have not found in the literature any other study where a similar analysis has been performed.

■

CONCLUSIONS This study reports detailed water sorption isotherms measured in a DVS instrument for four different polyion−surfactant ion complex salts and two reference simple surfactants in contact with a reservoir of gas hydrated in the range 0−95% RH. All complex salts show smooth sorption isotherms, most probably reflecting the dominance of the structurally similar hexagonal and rectangular phases across the studied RH range. Cycles of large stepwise changes in RH gave very reproducible sorption responses, showing that the complex salt films can be repeatedly hydrated/dehydrated with no change in performance. The equilibrium water uptake per unit mass for the studied complex salts increases with decreasing surfactant tail length and with decreasing polyion length, in accordance with previous findings for maximally swollen complex salts in equilibrium with excess water. The differences between C12 and C16 complex salts decrease when the hydration is expressed as the number of sorbed water molecules per ion pair. The latter hydration values are quite similar in magnitude to those previously found in sorption isotherms for well-studied ionomers such as Nafion, where the water uptake is also attributed to the ionic groups. A detailed kinetic model for the water exchange between the film and a surrounding gas reservoir is described and examined, where the central assumption is that water transport between the gas reservoir and the film surface (described in a stagnantlayer approach) is the rate-limiting process. A similar analysis of this limiting case seems not to have been presented previously. The most important predictions of the model are that, for a sufficiently small stepwise change in the reservoir RH, the water exchange should be a single-exponential process with a time constant τ that is proportional to both the slope K of the sorption isotherm over the interval in RH and the thickness of

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b02983. Graphs as in Figure 4 for C12TAPA6000, C16TAPA25, C12TAAc, and C16TAAc, graphs as in Figure 9 for C12TAPA6000, C16TAPA25, and C16TAPA6000, graphs showing τ/K ratios for the three different film thicknesses of each of the four CS, and table of time constants obtained for large step changes in RH as in Figure 5 (PDF)

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Olle Söderman, Emma Sparr, Stig Stenström, Lars Wadsö, and Håkan Wennerström for several helpful discussions and to Olle Söderman and Vitaly Kocherbitov for sharing data on sorption isotherms and self-diffusion coefficients prior to publication. This work was funded by the Swedish Foundation for Strategic Research (SSF), Grant RMA08-0056. 6789


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