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Ind. Eng. Chem. Res. 2011, 50, 580–589
Evaporation of Water from Structured Surfactant Solutions Paschalis Alexandridis,* Shushan Z. Munshi, and Zhiyong Gu† Department of Chemical and Biological Engineering, UniVersity at Buffalo, The State UniVersity of New York (SUNY), Buffalo, New York 14260-4200
Alkyl-propoxy-ethoxylate surfactant aqueous solution films are exposed to air of constant relative humidity, and the water loss from the film is monitored over time, until equilibrium is reached. The surfactants selfassemble into lyotropic liquid crystals with a structure that varies depending on the water concentration at a given time. The water loss data are analyzed to investigate the factors affecting the drying rate of the alkylpropoxy-ethoxylate surfactants, such as the air relative humidity, microstructure in the surfactant film, and the surfactant degree of hydration. Analytical solutions of the diffusion equation are used to extract the water diffusion coefficient in the film. A diffusion model that accounts for varying film thickness and evaporation at the surface has been used to follow the water loss and thickness data over time, in order to assess the relative importance of diffusion and evaporation under various conditions. Introduction Amphiphilic molecules, such as surfactants, lipids, and block copolymers, are able to self-assemble in various ordered structures in the presence of selective solvents.1,2 The structures attained by an amphiphile depend on a “preferred curvature”, which is initially set by the amphiphile molecular architecture but is also affected by the type and amount of solvent and solutes present.1-5 The spontaneous formation of supramolecular structures based on amphiphiles has attracted increasing interest due to its inherent beauty and its wealth of current and potential technological applications.1,2 The great majority of studies published in the open literature address systems at thermodynamic equilibrium. Yet, mass transport properties are of great importance in the utilization of the amphiphilic molecules.6 For example, mass transport is relevant in the surfactant dissolution process,7 and in drug delivery using carriers containing amphiphilic molecules.8-10 Transport properties such as evaporation rates are of interest in the assessment of hazards arising from volatile chemicals, in drying processes (e.g., during preparation of powders), and in the release of volatile active species such as perfumes and flavors from commercial products.11,12 Very few reports exist in the literature on the evaporation rate of water in structured surfactant solutions.11,12 Previous work in our group has examined the drying mechanism of hydrogels formed by poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEO-PPO-PEO) amphiphilic block copolymers.13,14 Block copolymers of the PEO-PPO-PEO type find numerous applications in industry and research due to the large variety of physicochemical properties that can be attained by varying the PEO/PPO ratio and relative block length, as well as the solution conditions.2,15-17 Lately, ABC-type triblock terpolymers have attracted attention due to their ability to adopt complex architectures.18,19 Following a comprehensive investigation on the thermodynamics, structure, and dynamics of the PEO-PPO type block copolymers,2,3,5,14-17 we turn our attention to ABCtype triblock terpolymers, in particular amphiphiles which consist of short PEO, PPO, and polyethylene (PE) blocks.20-23 The triblock architecture of these surfactants is expected to lead * To whom correspondence should be addressed. E-mail:
[email protected]. † Current Affiliation: Department of Chemical Engineering, University of Massachusetts, Lowell, MA 01854.
to an increased interaction spectrum, that is, a more complex hydrophilic/hydrophobic balance between the parts of the molecule that range from the very hydrophobic PE block to the hydrophilic PEO block through the moderately hydrophobic PPO block. Such surfactants with “graded” triblock composition are found useful in several applications.24-27 For example, nonyl phenyl-propoxy-ethoxylate surfactants with long PEO blocks have been used for the stabilization of aqueous dispersions.24 Surfactants of the alkyl-propoxy-ethoxy sulfate type have been used in enhanced oil recovery applications, where their graded nature offers an advantage in stabilizing oil-water interfaces over alkyl ethoxylates.25,26 In this study, we investigate the drying mechanism of alkylpropoxy-ethoxylate surfactant solutions by gravimetrically monitoring the water loss in air of controlled relative humidity (see schematic of Figure 1). In particular, we examine the effects of the drying environment, the surfactant hydrogel microstructure, and the surfactant “graded” composition. Analytical and numerical solutions of the diffusion equation are used to obtain the water diffusion in the surfactant hydrogel and to assess the relative importance of diffusion in the film and surface evaporation. This is one of very few studies on evaporation from structured surfactant solutions and the first study done on the transport properties of alkyl-propoxy-ethoxylate surfactants. Materials and Methods Materials. Three different alkyl-propoxy-ethoxylate surfactants with varying number, 8.5, 17, and 34, of ethylene oxide (EO) segments were used as received from Dow Chemical Company, Midland, MI (from here on, the surfactants will be denoted as “8.5 EO”, “17 EO”, and “34 EO”, respectively). These surfactants can be considered as short ABC triblock terpolymers, comprised of polyethylene, poly(propylene oxide),
Figure 1. Schematic of the aqueous surfactant solution film undergoing evaporation.
10.1021/ie100261u 2011 American Chemical Society Published on Web 07/28/2010
Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011
and poly(ethylene oxide) blocks. The “8.5 EO”, “17 EO”, and “34 EO” surfactants have molecular weights of 1266, 1641, and 2390 g/mol, respectively, and PEO contents of 30, 46, and 63 wt %, respectively. On the basis of their molecular weights and chemical composition, “8.5 EO”, “17 EO”, and “34 EO” can be represented by the formulas (C)13(PO)12.2(EO)8.5, (C)13(PO)12.2(EO)17, and (C)13(PO)12.2(EO)34, respectively (where C stands for methylene, PO for propylene oxide, and EO for ethylene oxide segment). The densities of the three surfactants are approximately 1.06 g/cm3. All salts and sample preparation method used in this study are the same as reported by Gu and Alexandridis.13,14 Drying Experiments. The driving force for the solvent (water) transfer from one phase to another is the difference in the solvent chemical potential between the different phases. In our study, the water chemical potential, ∆µ, at each air relative humidity condition examined is higher than that in the hydrogel sample (initially, 20 wt % surfactant in water), thus drying is achieved until equilibrium can be attained. ∆µ can be determined by the water vapor pressure in the air, p, as follows28,29 ∆µ ) RTln
()
p ) RTln(RH) p0
(1)
where p0 is the saturated water vapor pressure and RH is the relative humidity. Air of constant RH can be generated by saturated aqueous salt solutions.30 Saturated solutions of LiCl, NaBr, NaCl, KCl, KNO3, and K2SO4 were used to generate air RH of 10.9, 57.2, 75.3, 84.5, 93.8, and 97.4%, respectively. The methodology and the data acquisition method used in this study are the same as reported by Gu and Alexandridis.14 In brief, 20% aqueous surfactant solution was poured into a dish to form a film with an initial thickness of 5 mm (much smaller than the dish diameter so that the drying can be considered onedimensional). The samples were placed in a chamber containing still air of constant RH (maintained by saturated aqueous salt solutions), and the sample weight was monitored as a function of time. In order to minimize the disturbance to RH of opening the system (sealed dishes) to take measurements, the sealed dishes with the higher RH (75.3, 84.5, 93.8, and 97.4%) were placed in a Glovebag model X-37-37H (I2R, Cheltenham, PA) that maintained a high RH environment. For the lower RH cases (10.9 and 57.2%), the sealed dishes were kept in an airconditioned room with temperature at 24 °C ((1 °C) and ambient RH ranging from 30 to 60%. Results and Discussion Equilibrium Water Content: Hydration Isotherm. Depending on the relative humidity, the equilibrium water concentration of the hydrogel is different and corresponds to different ordered structures. The physical and chemical stability of the surfactant is influenced by the water activity (and correspondingly by the air RH). Hydration isotherms are of great relevance to the drying experiments because they assist in setting the end-point of the drying process and in determining the packaging and storage requirements for a desired shelf life.31 The weight of the hydrogel that has been measured gravimetrically was converted into water concentration in weight percent and is presented in the form of an adsorption isotherm in Figure 2. The water concentration values reported are the average values over the whole film. At lower RH, the surfactant hydrogel retains very low water content. As the air RH increases, the equilibrium water concentration in the hydrogel increases.
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Figure 2. Equilibrium water concentration (wt %) as a function of water vapor activity (P/P0) for (O) “8.5 EO”, (4) “17 EO”, and (0) “34 EO” surfactants.
Different ordered structures are formed at different relative humidity conditions. At the experiment temperature (24 °C), the hexagonal structure forms at RH approaching 100%, lamellar structure at 94-97% RH, and high-polymer-content phase at 8-84% RH for the “8.5 EO” hydrogel. For the “17 EO” hydrogel, the micellar cubic phase forms at 100% RH, the hexagonal phase at 97% RH, the lamellar phase at 75-94% RH, and the high-polymer-content phase at 8-64% RH. The “34 EO” hydrogel exhibits micellar cubic phase at 97-100% RH, hexagonal phase at 85-94% RH, bicontinuous cubic phase at 81% RH, and semicrystalline pastelike phase at 8-75% RH. The above correspondence between relative humidity and structure is based on separate studies of the equilibrium phase behavior and structure as a function of composition (fixed water content). The phase boundaries thus determined at equilibrium are denoted with dotted lines in Figure 3. While some hysteresis may be possible during drying, when the local composition changes drive a phase change, the time scale of the drying experiments is much greater than the time scale of phase transitions in similar surfactant lyotropic liquid crystalline systems. During drying, the chemical and physical properties change,14,32 and thus it is our interest to investigate the water transport properties within the surfactant hydrogel until equilibrium is reached. Time Evolution of Water Loss. The hydration isotherms presented above establish the water content of the hydrogel films in contact with air for different RH conditions at equilibrium (infinite time). In order to understand the solvent (water in this case) transport process, we monitored the kinetics of water loss from the hydrogel film. Figure 3 shows the evolution of the alkyl-propoxy-ethoxylate surfactant concentration (average over the whole film) as a function of drying time at the 10.9, 57.2, 75.3, 84.5, 93.8, and 97.4% air RH. For all three surfactants considered here, at the very start of the drying process, a slow increase of the surfactant concentration is observed, followed by a rapid increase. As the surfactant hydrogels approached their equilibrium concentration, the increase slows down and then reaches a plateau, where the surfactant concentration did not change further. As expected, the equilibrium is reached faster at the lower RH conditions. The total water weight loss at any given time is defined as the difference between the water weight of the hydrogel film at the start of the drying process and the water weight of the hydrogel at any given time. The water weight loss data shown
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Figure 3. Average surfactant concentration in the film as a function of drying time at air RH of (O) 10.9%, (4) 57.2%, (0) 75.3%, (]) 84.5%, (3) 93.8%, and (right pointing open triangle) 97.4% for the (a) “8.5 EO”, (b) “17 EO”, and (c) “34 EO” surfactants. The dotted lines represent phase boundaries at 24 °C. L1 f homogeneous liquid solution; I1 f micellar cubic lyotropic liquid crystal; H1 f hexagonal lyotropic liquid crystal; V1 f bicontinuous cubic lyotropic liquid crystal; LR f lamellar lyotropic liquid crystal; L2 f high-polymer content solution.
in Figure 4 are calculated as water weight lost per surface area of the hydrogel film exposed to the air (which is a constant value of 9.62 × 10-4 m2 for our experiments). The weight of water lost from the hydrogel film increases initially linearly with the drying time, followed by a nonlinear increase, until it reaches a constant value. A two-stage mechanism13,33,34 can be discerned in the water weight loss curve: the linear region of water weight loss versus time corresponds to stage I, and the nonlinear region corresponds to stage II of the drying. The two-stage mechanism
Figure 4. Total water loss (gram/square meters) at any given time, plotted as a function of drying time at air RH of (O) 10.9%, (4) 57.2%, (0) 75.3%, (]) 84.5%, (3) 93.8%, and (right pointing open triangle) 97.4% for the (a) “8.5 EO”, (b) “17 EO”, and (c) “34 EO” surfactants. The solid lines represent the numerical solution to the diffusion model for the air RH conditions of 10.9-84.5%.
is observed for all three surfactants at all the RH conditions considered here. Drying Rate. The drying rate is defined as the derivative of water loss with respect to drying time. The drying rate is constant at the initial stages of drying (stage I), followed by a drop (stage II of the drying process). This constant drying rate at each air RH is plotted in Figure 5. A linear trend is observed when the drying rate at stage I is plotted as a function of the air relative humidity for all three surfactants. The drying rate drops by about 1/2 as the air RH increased from 10.9% to 57.2% RH.
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Figure 5. Drying rate of stage I as a function of air RH for the (O) “8.5 EO”, (4) “17 EO”, and (0) “34 EO” surfactants.
Stage I of the drying process, also referred to as the constantrate period, is the first major drying period.35 Since the initial water content in the surfactant hydrogel film is very high (80 wt %), the surfactant film is a homogeneous liquid solution where water molecules can move relatively freely. Thus, the drying process in stage I is limited by the water evaporation from the film surface.14,35 As shown in Figure 5, the external condition (i.e., air RH) is the primary factor affecting the water evaporation rate of the surfactant hydrogel, since the drying rates of all three surfactants at state I almost overlap with each other at each air RH. As the drying proceeds, the morphology of the hydrogel film starts to change and both the surface water evaporation and the water transport inside the structured hydrogel would affect the drying rate, which corresponds to stage II of drying.14 A two stage drying mechanism was observed in the drying of hydrogel films formed by Pluronic P105 and F127 PEOPPO-PEO block copolymers13,14 and in other polymer solutions.34 A linear drop in the drying rate at stage I as the air RH increases has also been observed in the Pluronic F127-water and Pluronic P105-water systems.14 Structural Effects on Drying Rate. Studies have shown that the permeability or molecule diffusion is affected by ordered morphologies formed by block copolymers36 or semicrystalline polymers.37 The microstructure of the three alkyl-propoxyethoxylate surfactants considered here may have an influence on the drying rate. To examine this, we plot in Figure 6 the drying rate as a function of the water content, which spans several phases (and corresponding microstructure). For the case of the “8.5 EO”-water system (Figure 6a), the drying rate remains constant in the homogeneous liquid solution and the hexagonal phase and decreases starting from the lamellar phase until the high-polymer-content phase, where a greater drop in the drying rate is observed. The same trend is observed in the case of the “17 EO”-water system (Figure 6b); however, the decrease in the lamellar phase is steeper than that in “8.5 EO”water system. In the case of “34 EO”-water system, the drying rate remains constant in the homogeneous liquid solution and the micellar cubic phase and decreases starting from the hexagonal phase. To compare the surfactants at the same air RH, the drying rates for all three surfactants at 57.2 and 84.5% RH are plotted as a function of water content in parts a and b of Figure 7, respectively. At the initial stages of drying (high water content), the drying rates of the three surfactants are very similar. This
Figure 6. Drying rate as a function of water content at air RH of (O) 10.9%, (4) 57.2%, (0) 75.3%, (]) 84.5%, (3) 93.8%, and (right pointing open triangle) 97.4% for the (a) “8.5 EO”, (b) “17 EO”, and (c) “34 EO” surfactants. The dotted lines represent the phase boundaries. The notation used for the different phases is the same as in Figure 3.
is consistent with our finding that the initial stages of the drying process are determined by the air RH, regardless of the surfactant type. In the intermediate stages of the drying, a trend is observed where the “8.5 EO” surfactant system has the highest drying rate, followed by the “17 EO” system, and then the “34 EO” system. As shown in the hydration isotherm (Figure 2), there is higher water content at the end of the drying process for the higher PEO surfactant, and therefore there is less water diffusing out of the hydrogel film and a lower drying rate. Studies on PEO-PPO-PEO block copolymers have shown that the drying rate is lower when the PEO content of the block copolymer is higher.14
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Figure 7. Drying rate as a function of (a, b) water content, (c, d) “corrected” water content when assuming the hydrophobic domain to consist of both the alkyl part and the PPO part, and (e, f) when assuming the hydrophobic domain to consist of only the alkyl chain, at (left panels) 57.2% RH and (right panels) 84.5% RH, for the (O) “8.5 EO”, (4) “17 EO”, and (0) “34 EO” surfactants.
As mentioned previously, the surfactant self-assembled structure may be a factor in determining the drying rate. In parts a and b of Figure 7, the drying rate for all three surfactants starts to decrease at approximately the same water content, implying that the water content or the hydration level of the surfactant may be important. To further examine this point, we tried to decouple the self-assembled structure from the hydration level. The intermolecular interactions in surfactant hydrogels originate mainly from interactions between the hydrophilic segments and water.38 Taking this into account, we attempt to “correct” the water content by factoring out the hydrophobic domains. Water is a good solvent for poly(ethylene oxide),
intermediate solvent for poly(propylene oxide), and a bad solvent for the alkyl chain.21 Therefore the hydrophobic domains can be considered to consist of either only alkyl chains or both the alkyl and PPO parts of the surfactant. The “corrected” water content, H2O*, when assuming the hydrophobic domains to consist of the alkyl and the PPO parts is defined as14 H2O* )
H2O wt % (2) (100 - H2O wt %)PEO wt % + H2O wt %
and when assuming hydrophobic domains to consist of only the alkyl chains is defined as
Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 H2O ) *
H2O wt % (3) (100 - H2O wt %)(PPO wt % + PEO wt %) + H2O wt %
where H2O wt % is the water content in the surfactant hydrogel and PEO wt % and PPO wt % are the PEO and PPO contents of the surfactants, respectively. The drying rates at 57.2% and 84.5% RH are plotted as a function of the hydration level or “corrected” water content when assuming the hydrophobic domains to consist of the alkyl and the PPO parts in Figure 7c,d and when assuming hydrophobic domains to consist of only the alkyl chains in Figure 7e,f. If the hydration level were to play a more important role than the surfactant microstructure in the hydrogel film, then the drying rates of all three surfactants should overlap when plotted as a function of the “corrected” water content. Indeed, the drying rates at 57.2 and 84.5% RH overlap fairly well when the hydrophobic domains is taken to consist of both alkyl and PPO parts (Figure 7c,d), indicating that the hydration level does affect the drying rate. When the hydrophobic domains are assumed to consist of only the alkyl chains, then no overlap is observed (Figure 7e,f). This indicates that PPO is acting as hydrophobic rather than as hydrophilic. Studies on drying rates of PEO-PPO-PEO block copolymers and PEO homopolymers have also shown that the hydration level rather than the microstructure is the determining factor for the drying rate (note that the PEO-PPO-PEO drying rate data points overlapped very well with the PEO homopolymer data points).14 A similar approach was used when plotting the osmotic pressure as a function of the PEO concentration, which is calculated based on the block copolymer concentration and PEO content in the block copolymer molecules.38 The osmotic pressure data points for the PEO-PPO-PEO block copolymers overlapped very well with the PEO homopolymer data points. This indicates that the osmotic pressure originates mainly from interactions between PEO and water.38 The observation that the microstructure does not play a dominant role in the transport properties possibly emanates from the water-continuous topology in the systems considered. Even in the anisotropic lamellar phase, structural defects may allow the movement of water perpendicular to the (hydrophobic) surfactant layers. Water Diffusion (Analytical Solutions for Limiting Cases). As mentioned previously, the drying rate is affected by water transport (diffusion) inside the structured hydrogel. The water diffusion can be quantified by extracting the water diffusion coefficient from the experimental water loss data. The (one-dimensional) water transport in the surfactant hydrogel can be described by Fick’s second law:39 ∂C ∂ ∂C ) D ∂t ∂x ∂x
( )
(4)
where t is the drying time, x is the direction of the water diffusion (normal to the surface of the film), D is the diffusion coefficient of water, and C is the water concentration in the surfactant hydrogel film. The water diffusion coefficient can be obtained using an analytical solution of eq 4.13,39,40 Several assumptions are made to this end: (i) the drying process is onedimensional, (ii) the drying process is isothermal, (iii) the film thickness is constant, (iv) the water diffusion coefficient is constant, and (v) the concentration at the surface of the hydrogel film is the same as inside the film. On the basis of these initial/ boundary conditions and assumptions, a trigonometric series analytical solution of eq 4 can be obtained. Long time and short
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time expansions of this trigonometric series lead to eqs 5 and 6, respectively.40,41
(
∞
Mt )1M∞
∑ (2n +8 1) π
2 2
exp
n)0
Mt 4 Dt ) M∞ l π
( )
0.5
)
(5)
)
(6)
-(2n + 1)2π2Dt l2
(
∞
∑
nl 8 + (Dt)0.5 (-1)n ierfc l 2(Dt)0.5 n)1
where Mt is the water lost per unit area of film at time t, M∞ is the maximum water loss, and l is the film thickness. Equation 5 converges rapidly at long times, whereas eq 6 converges rapidly at short times.40 In what follows we present methods which are based on eq 5 or 6 for extracting a diffusion coefficient from water vapor sorption/drying data. Method 1. One method to estimate the water diffusion coefficient is from the measurement of the sorption/drying halftime, t0.5, obtained from a two-term approximation of eq 5:40,41 t0.5 )
0.04919l2 Dh
(7)
where the sorption/drying half-time is defined as the time where Mt/M∞ ) 0.5. Method 2. At short times, (Mt/M∞ < 0.5), the following expression can be obtained from the first term of eq 6:40,41
( )
Mt 4 DIt ) M∞ l π
0.5
(8)
The water diffusion coefficient (DI) can be calculated from the initial slope of a plot of Mt/M∞ vs t0.5. Method 3. At longer times (Mt/M∞ > 0.5), the following expression can be obtained from the one-term approximation of eq 5:40,41
(
ln 1 -
)
Mt π2DFt 8 ) ln 2 M∞ π l2
(9)
The water diffusion coefficient (DF) can be calculated from the limiting slope of ln(1 - Mt/M∞) vs t. Method 2 was used for fitting data sets at short times, whereas method 3 was used for fitting data sets at longer times. The fitting of the analytical solution to the drying data using method 2 is shown in Figure 8. The water diffusion coefficient (DI) values of all three surfactants for 93.8% and 97.4% RH were calculated based on the M∞ from the equilibrium data (Figure 2). The water diffusion coefficients at 10.9%, 57.2%, 75.3%, and 84.5% RH for all three surfactants using methods 1, 2, and 3 were calculated based on the kinetics data (Figure 3). When using method 2, the first few data points were omitted because they were at the initial stages of the drying process, and when using method 3, the last few data points were omitted because they were close to the equilibrium states. Table 1 reports the water diffusion coefficient values extracted via eqs 7-9. The diffusion coefficient obtained for the “8.5 EO”, “17 EO”, and “34 EO” systems at each air relative humidity from methods 1, 2, and 3 were fairly similar. With the comparison of the diffusion coefficients values of a given surfactant at varying RH, a general trend is that the diffusion coefficient decreases as the air relative humidity increases. When the three surfactants are compared at the same RH, the diffusion coefficient values obtained from methods 1 and 2 were very similar for all three surfactants at each air RH.
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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 Table 1. Water Diffusion Coefficients in Surfactant Hydrogels Obtained from Analytical Solutions Diffusion coefficient (m2/s) surfactant
air RH (%)
Dha
DIb
DFc
“8.5 EO”
10.9 57.2 75.3 84.5 93.8 97.4 10.9 57.2 75.3 84.5 93.8 97.4 10.9 57.2 75.3 84.5 93.8 97.4
4.5 × 10-12 2.2 × 10-12 1.3 × 10-12 7.6 × 10-13
1.1 × 10-11 4.2 × 10-12 2.1 × 10-12 1.5 × 10-12 5.6 × 10-13 2.0 × 10-13 1.0 × 10-11 4.5 × 10-12 3.0 × 10-12 1.5 × 10-12 4.8 × 10-13 2.6 × 10-13 1.1 × 10-11 4.0 × 10-12 2.8 × 10-12 1.3 × 10-12 5.1 × 10-13 1.1 × 10-13
1.4 × 10-10 1.0 × 10-10 2.4 × 10-11 5.9 × 10-12
“17 EO”
“34 EO”
4.7 × 10-12 2.1 × 10-12 1.2 × 10-12 7.7 × 10-13 4.7 × 10-12 1.9 × 10-12 1.2 × 10-12 8.1 × 10-13
1.4 × 10-10 3.9 × 10-11 8.6 × 10-12 5.7 × 10-12 1.0 × 10-11 7.8 × 10-12 7.7 × 10-12 5.0 × 10-12
a Value extracted from eq 7. b Value obtained from fitting eq 8 to the experimental results at shorter times. c Value obtained from fitting eq 9 to the experimental results at longer times.
2.3 × 10-12 m2/s at 94% RH when calculated using methods 1, 2, and 3.41 The order of magnitude of these diffusion coefficients is comparable to the findings of the present study. The utilization of analytical solutions discussed above has merit on the basis of the simplicity in calculation and also as a means of comparison of our study to several published studies that utilize such analytical solutions. A numerical solution allows the use of transport equations that are a more accurate representation of physical phenomena, thus we have used such approach in our work. Water Diffusion (Numerical Solution). To better analyze the evaporation process, we solved the diffusion equation with a numerical model that accounts for the water diffusion in the film and varying film thickness. If we consider the water vapor pressure in contact with the film to be constant at all times and the film thickness to decrease with time (i.e., moving boundary condition), the boundary condition at the air-film interface is obtained by using the jump mass balance:14,42 -D
Figure 8. Fitting of the analytical solution (eq 8) at data obtained for air RH of (O) 10.9%, (4) 57.2%, (0) 75.3%, (]) 84.5%, (3) 93.8%, and (right facing open triangle) 97.4% for the (a) “8.5 EO”, (b) “17 EO”, and (c) “34 EO” surfactants. DI represents the diffusion coefficient value (meters squared/second) obtained at each RH.
In the case of diffusion coefficient values obtained from method 3, a trend is observed where “8.5 EO” has the highest value, followed by “17 EO”, and the lowest value was for “34 EO”. Method 3 is calculation based on longer time scales, where the three surfactants possess different microstructures. Therefore, a difference in the apparent diffusion coefficient is observed. The methods mentioned above were used in obtaining the water diffusion coefficient in other systems such as Pluronic P105.41 For this PEO-PPO-PEO block copolymer, the diffusion coefficients were in the range 4.8 × 10-12 to 7.3 × 10-12 m2/s at 38% RH, 2.9 × 10-12 to 5.2 × 10-12 m2/s at 58% RH, 1.2 × 10-12 to 2.4 × 10-12 m2/s at 85% RH, and 1.3 × 10-12 to
dh(t) ∂C -C ) R(C - C∞), ∂t dt
at x ) h(t)
(10)
where h(t) is the film thickness, the term C(dh(t)/dt) accounts for the water flux due to the moving boundary, and R is a constant that represents the product of the mass transfer coefficient and the proportionality constant.14 The boundary condition at the hydrogel film-dish interface is D
∂C ) 0, ∂x
at x ) 0
(11)
The initial condition is C ) C0,
at t ) 0 for 0 > x > L0
(12)
where C0 is the initial water concentration and L0 is the initial film thickness. The water diffusion coefficient is considered to be a function of water concentration in the hydrogel film:14,43 D ) D0e-k/C
(13)
where D0 is the pre-exponential factor and k is a constant with the same units as the water concentration. This model (eq 4
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Table 2. Parameters Obtained from Fitting the Diffusion Model to Experimental Water Loss Data R (m/s)
Figure 9. Film thickness as a function of drying time at air RH of (O) 10.9%, (4) 57.2%, (0) 75.3%, (]) 84.5%, (3) 93.8%, and (right facing open triangle) 97.4% for the (a) “8.5 EO”, (b) “17 EO”, and (c) “34 EO” surfactants. The solid lines represent the numerical solution to the diffusion model for the air RH conditions of 10.9-84.5%.
with the boundary conditions of eqs 10 and 11) was solved numerically to fit the experimental data in order to extract the water diffusion coefficient and the mass transfer coefficient. The water loss as a function of time data were fitted using this numerical model, and the film thickness as a function of time was predicted using the parameters obtained from the fitting. The solid lines shown in Figures 4 and 9 are the fitting results for water loss and film thickness, respectively, obtained from the numerical solution. Table 2 summarizes the parameters obtained from the numerical fit. Three parameters (D0, R, and k) are needed for the numerical model to predict the water loss data and film thickness time
surfactant
10.9% RH
57.2% RH
75.3% RH
84.5% RH
“8.5 EO” “17 EO” “34 EO”
-9
-9
-9
1.7 × 10-9 1.8 × 10-9 2.2 × 10-9
9.0 × 10 9.0 × 10-9 9.0 × 10-9
4.5 × 10 4.4 × 10-9 3.8 × 10-9
2.6 × 10 2.6 × 10-9 2.8 × 10-9
evolution. D0 was set at the self-diffusion coefficient of neat water, 2.23 × 10-9 m2/s at 25 °C,44 and k was set (after several trials) to an optimum value of 0.1. The R impacted more on the fitting of the initial stages of the drying process (approximately 1/3 of the total drying time). As mentioned earlier, the initial stage of the drying mechanism is governed by evaporation at the surface of the film, and this can be confirmed by observing the R values (R is directly related to the mass transfer coefficient14) found in Table 2. For all surfactants considered here, R decreases as the air RH increases. The R values are the same (or very similar) for the three surfactants at each air RH. A similar trend of R being a function of air RH has also been observed in the PEO-PPO-PEO block copolymerwater systems.13,14 The mechanism of solvent drying from amorphous polymers, such as benzene,45,46 ethyl acetate,45 ethyl methyl ketone,47 and ethyl benzene48 from polystyrene, has been reported. In these systems, the drying curve resembles a viscoelastic (non-Fickian) diffusion curve. For example, the drying of ethyl methyl ketone in atactic polystyrene exhibited a sequence of “sigmoid” type f “pseudo-Fickian” type f “two-stage” type f “pseudoFickian”type f ”Fickian” type characteristics.47 The term “sigmoid” describes an S-shaped behavior of solvent loss versus square root of time. “Pseudo-Fickian” describes a linear increase of solvent loss with the square root of time (as predicted by Fick’s law) with a prolonged approach to equilibrium. The “twostage” behavior is when the solvent loss appears to reach an equilibrium level rather quickly but subsequently relaxes upward from the equilibrium value over a long time scale.48 A similar viscoelastic drying mechanism is observed for the three surfactant systems considered here, where the water loss curves can be described as following a “sigmoid” f “pseudo-Fickian” f “two-stage” behavior. The Fickian diffusion model and the experimental data overlap very well at the initial stages of the drying, in particular at the “sigmoid” and “pseudo-Fickian” regions. However, the numerical solution does not capture the water loss data as the surfactant solution approaches equilibrium (“two-stage” behavior region), indicating that viscoelastic-type effects are relevant in this region. The Biot number, Bi, often used in heat transfer where it represents the ratio of conduction to convection heat resistance, is also relevant to drying processes. In this case, the Biot number represents the ratio of the internal to external mass transfer resistance:49,50 Bi )
RL0 D0
(14)
where L0 is the initial surfactant hydrogel film thickness and D0 is the pre-exponential factor of the water diffusion coefficient. When Bi , 1, drying is limited by external mass transfer resistance, which is the water evaporation at the surface of the surfactant hydrogel. When Bi . 1, drying is limited by internal mass transfer resistance, which is the water diffusion in the hydrogel film. When Bi ≈ 1, drying is dictated by both external and internal resistance.49,50
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The Biot numbers were found to be on average 0.0202 at 10.9% RH, 0.0095 at 57.2% RH, 0.0060 at 75.3% RH, and 0.0042 at 84.5% RH. The Biot number is less than unity for all three surfactant hydrogel films, indicating that drying is limited by the water evaporation at the surface. Note that these findings correspond to early stage drying (stage I). When the drying proceeds, especially when it reaches stage II, both evaporation and water diffusion have an effect and the Bi number is expected to increase to a relatively larger number (L0 will decrease but D0 will decrease even further). Concluding Remarks The drying of aqueous solutions of three alkyl-propoxyethoxylated surfactants with varying PEO content has been investigated here. Films of surfactant solutions were exposed to air of known water vapor pressure until equilibrium was achieved and the drying kinetics monitored. A two stage drying mechanism with a constant drying rate region (stage I) followed by a drop in drying rate (stage II) was observed. The drying rate at stage I was a linear function of the air relative humidity but depended very little on the surfactant type. These novel alkyl-propoxyl-ethoxylate surfactants are capable of self-assembling into a variety of lyotropic liquid crystal ordered structures, depending on the water content. To probe the effect of surfactant microstructure on the drying rate, the drying rate was plotted as a function of water content. The drying rate started decreasing at approximately the same water content for all three surfactants, suggesting that hydration level dictates the drying rate. The hydration effect was further analyzed by plotting the drying rate as a function of the “corrected” water content. The data for the three surfactants overlapped very well when plotted as a function of the “corrected” water content indicating that the hydration level determines the drying rate when assuming that the hydrophobic domain consists of both the alkyl and the PPO parts. In addition to the importance of the hydration level, we can also conclude that the PPO part of the “graded” alkyl-propoxy-ethoxylate surfactants acts more hydrophobic than hydrophilic. Analytical solutions of Fick’s second law that describe the water diffusion in the surfactant hydrogel have been fitted to the experimental water loss data in order to extract the water diffusion coefficient. The water diffusion coefficient values calculated from the half time, short time scales, and longer time scales of the drying process are comparable. The diffusion equation was solved numerically to better assess the evaporation at the high water content region. The mass transfer coefficient extracted from fitting the experimental data with the numerical method established that the initial stages of the drying are controlled by the drying conditions (i.e., the air RH). Acknowledgment Financial support from the National Science Foundation (NSF) is greatly appreciated. We thank Dr. S. Balijepalli and Dr. H. J. M. Gruenbauer of Dow Chemical Co. for providing the surfactants used in this study. Literature Cited (1) Evans, D. F.; Wennerstrom, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet; VCH Publishers: New York, 1994. (2) Alexandridis, P.; Lindman, B. Amphiphilic Block Copolymers: SelfAssembly and Applications; Elsevier Science B. V.: Amsterdam, The Netherlands, 2000.
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ReceiVed for reView February 2, 2010 ReVised manuscript receiVed July 1, 2010 Accepted July 12, 2010 IE100261U
