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Langmuir 2004, 20, 9560-9564
Diffusion of Molecules Confined in Semipenetrable Nanoscale Carriers Probed by Pulsed Field Gradient NMR Jaroslav Krˇ´ızˇ* Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovsky Sq. 2, 162 06 Prague 6, Czech Republic Received May 27, 2004. In Final Form: August 17, 2004 Diffusion of three low-molecular-weight compounds cyclohexane (CX), benzene (BZ), and chloroform (CL) preferentially confined in the cores of nanoscale carriers was probed by pulsed field gradient (PFG) NMR methods. The carriers were monolayer micelles of sodium dodecyl sulfate (SDS), bilayer micelles of poly(hexyl methacrylate)-block-(acrylic acid) (M2), and trilayer micelles of poly(2-ethylhexyl methacrylate)block-(methyl methacrylate)-block-(acrylic acid) (M3) in D2O at 300 K. Although the radius of the confinement space was 10-8 m or lower, the course of the PFG signal attenuation in pulsed gradient spin-echo or stimulated echo experiments under varied diffusion time corresponds to apparently unrestricted diffusion, which is slowed down compared to that of the compound dissolved in D2O. Analysis using approximate relations reveals that the response of the system to PFG NMR consists of three independent components, namely (i) diffusion of the carrier as a whole, (ii) hindered escape of a confined molecule and its diffusion in the medium, and (iii) diffusion of the molecules dissolved in the medium. If process ii is fast enough, exchange of the compound between the carrier and the medium includes the influence of iii as a component of a monoexponential PFG decay; otherwise, two sets of signals are observed with different diffusion responses, or biexponential PFG is observed. According to the results of this study, the only barrier of the diffusion of the inspected compounds CX, BZ, and CL out of their confinement in the carriers SDS or M2 is a thermodynamic one, that is, the resistance of the saturated solution to accept surplus molecules of the solute. In a three-layer micelle M3, the additional polymer sheet around the confinement area forms an additional diffusion barrier for CX, however. The study shows that PFG NMR, though unable to observe directly restricted diffusion on the nanoscale, can be useful in studying systems designed, for example, for a controlled release of low-molecular-weight substances.
Introduction Micelles of detergents and amphiphilic block copolymers have been studied1-7 for their potential applicability as carriers of various active substances. For the controlled release of, for example, drugs confined in their cores, information about their diffusion mobility is of primary interest. In our previous NMR studies of solubilization of various substances by polymer micelles,8-10 this information was inferred from the shape, position, and intensity of 1H NMR signals of the solubilized substance diffusing freely into the micelles through surrounding water.9 Pulsed field gradient (PFG) NMR should offer direct information about self-diffusion mobility of substances either in homogeneous11,12 or in highly inhomogeneous, * E-mail:
[email protected]. Phone: +420-296809410.
+420-296809382. Fax:
(1) Tuzar, Z.; Webber, S. E.; Ramireddy, C.; Munk, P. Polym. Prepr. (Am. Chem. Soc., Div. Polym. Chem.) 1991, 32, 525. (2) Hurter, P. N.; Hatton, T. A. Langmuir 1992, 8, 1291. (3) Kiserow, D.; Procha´zka, K.; Ramireddy, C.; Tuzar, Z.; Munk, P.; Webber, S. E. Macromolecules 1992, 25, 4613. (4) Zhang, L.; Einsenberg, A. Science 1995, 268, 1728. (5) Chu, B. Langmuir 1995, 11, 414. (6) Moffit, M.; Zhang, L.; Khongaz, K.; Eisenberg, A. In Solvents and Selforganization of Polymers; Webber, S. E., Munk, P., Tuzar, Z., Eds.; Kluwer Academic Publishers: Dodrecht, 1996; p 53. (7) Almgren, M.; Brown, W.; Hewidt, S. Colloid Polym. Sci. 1995, 273, 2. (8) Krˇ´ızˇ, J.; Masarˇ, B.; Pospı´sˇil, H.; Plesˇtil, J.; Tuzar, Z.; Kiselev, M. A. Macromolecules 1996, 29, 7853. (9) Krˇ´ızˇ, J.; Masarˇ, B.; Doskocˇilova´, D. Macromolecules 1997, 30, 4391. (10) Krˇ´ızˇ, J.; Masarˇ, B.; Plesˇtil, J.; Tuzar, Z.; Pospı´sˇil, H.; Doskocˇilova´,D. Macromolecules 1998, 31, 41. (11) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (12) Tanner, J. E. J. Chem. Phys. 1970, 52, 2523.
for example, porous,13-15 media. However, the use of PFG NMR in the case of semipermeable organic micelles is not as straightforward as one could expect. It is the task of this short study to probe this use in several practically interesting cases. Experimental Part Materials and Sample Preparation. Cyclohexane (CX), benzene (BZ), chloroform (CL), and sodium dodecyl sulfate (SDS) were all analytic grade (Aldrich) and were used as obtained. SDS was dissolved in D2O to a 3 wt % solution. Block copolymers poly(hexyl methacrylate)-block-(acrylic acid) (M2) and poly(2ethylhexyl methacrylate)-block-(methyl methacrylate)-block(acrylic acid) (M3) were prepared from the corresponding monomer by group-transfer polymerization.10 They were micellized by transferring them from a tetrahydrofuran solution into water by dialysis, after which water was removed by vacuum lyofilization. The lyofilizate was then dissolved in D2O (99.9% d) to a 0.5 wt % solution. A total of 1 mL of the micellar solution was mixed with 0.5 mL of the corresponding substance and vigorously stirred in a closed vessel for 4 h at 300 K (SDS), 24 h at 330 K (M2), and 4 days at 330 K (M3). The resulting mixtures were then centrifuged at 3600 rpm for 4 h, and the aqueous layer was then transferred into a NMR tube. These conditions were checked by 1H NMR as sufficient for reaching equilibrium (except the system of CX with M3, where true equilibrium was not reached). PFG NMR Measurements. All NMR measurements were done at 300.13 MHz with a Bruker Avance DPX 300 spectrometer upgraded by two z-gradient-producing systems, namely, 0-50 G/cm with a conventional inverse-detection probe (used for high (13) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Oxford University Press: New York, 1991. (14) Stepisnik, J. Physica B 1993, 183, 343. (15) Wang, L. Z.; Caprihan, A.; Fukushima, E. J. Magn. Reson., Ser. A 1995, 117, 209.
10.1021/la048690y CCC: $27.50 © 2004 American Chemical Society Published on Web 09/29/2004
Molecule Diffusion in Semipenetrable Carriers
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sensitivity, in a direct-detection mode) and 0-1200 G/cm with a water-cooled diffusion probe. All measurements were done with an internal deuterium lock. Pulsed gradient spin-echo (PGSE)11 and pulsed gradient stimulated echo (PGSTE)12 methods were used with 1-ms gradient pulses. Gradient values were incremented linearly in 16 or 32 steps. The number of scans was from 16 to 160 according to sensitivity, the relaxation delay was 5T1, usually 30 s, and the diffusion delay (held constant in one experiment) was varied from 60 to 220 ms. The free induction decays were Fourier transformed in a phase-sensitive mode, and the integral intensities of the selected signals were fitted to the expected exponential decay function using a least-squares algorithm provided by the spectrometer’s software. For the measurement of the partition coefficient ξcs of the solute between the given micelle and D2O defined as ξcs ) φc/1 - φc, φc being the weight fraction of the solute in the core, integral intensities of the NMR signal of the given substance relative to the signals of the micellar carrier were used, corrected by the solubility of the substance in D2O obtained in a separate experiment.
Results and Discussion Following the analysis by Fukushima et al.,15 the dephasing of spin vectors and the resulting NMR signal attenuation rests on the probability of transition of a given spin from the position ri at the time ti to rj at tj. This is given by the propagator P(ri,ti|rj,tj) following the FokkerPlanck equation
∂P(ri,ti|rj,tj) ∂(ti - tj)
) D∇2P(ri,ti|rj,tj)
(1)
This obvious analogy to the second Fick’s law implicitly assumes a constant diffusion coefficient D. Otherwise, eq 1 should read
∂P(ri,ti|rj,tj) ∂(ti - tj)
) ∇D(r) ∇P(ri,ti|rj,tj)
(1a)
provided that D(r) is differentiable. With a being the mean path length of unrestricted diffusion in the direction of the field gradient, ∆ the diffusion delay used in a PGSE or PGSTE experiment, and δ the length of the gradient pulse with a magnitude g, three main regimes of the attenuation of the signal intensity S(g) by spin dephasing due to diffusion can be discerned.15 Using Fukushima’s terms, we can have rapid, restricted, and unrestricted diffusion according to the relation between diffusion mobility and the size of the area or the length of the gradient pulse. In the following, we will consider the type of PFG experiment in which ∆ and δ are held constant and g is varied so that S(g) is not influenced by relaxation. If the restriction area is spherical, the approximate relations of signal attenuation are
(i) rapid diffusion (δD . a2) S(g) ≈ exp(-16a4γ2g2δ/175D)
(2)
(ii) restricted diffusion (∆D . a2 but δD , a2) S(g) ≈ exp(-a2γ2g2δ2/5)
(3)
(iii) unrestricted diffusion (∆D , a2) S(g) ) exp[-γ2g2δ2D(∆ - δ/3)] ≈ exp[-γ2g2δ2D∆] (4) Taking micellar cores as an example, we usually have a e 10-8 m. In standard PFG experiments, δ ≈ 10-3 s, and ∆ is of the order of 10-2 s. For low-molecular-weight
Table 1. Solubility s (g/L) and Self-Diffusion Coefficient Ds (m2 s-1) in D2O at 300 K solute
s
Ds × 109
CX BZ
0.05 1.73
0.84 1.02
solute CL
s
Ds × 109
7.84
1.07
substances, 10-10 < D e 10-9 m2 s-1. If the core walls are truly impenetrable and the micelle as a whole does not move, we should be squarely in the regime i: there would be almost no signal attenuation with realistic gradient values up to 10 G m-1. Nevertheless, let us consider other regimes, too. Comparing eq 2 to eq 4, we can see that the experimental dependence of S(g) on g with constant ∆ and δ in all cases is
S(g) ) exp[-ζg2]
(5)
where ζ is independent of ∆ in regimes i and ii and linearly dependent on ∆ in iii, whereas its dependence on δ is linear in i and quadratic in ii and iii. Thus, a series of experiments with varied ∆ and δ should give the type of diffusion restriction in the given system. This is checked below with three differently watersoluble low-molecular-weight diffusants, namely, CL, BZ, and CX. Table 1 gives their values of solubility and selfdiffusion coefficients in D2O at 300 K. The systems into which these compounds were absorbed were: (a) SDS micelles (3 wt % in D2O); (b) micelles of M2 (0.5 wt % in D2O, radii of the core and whole micelle 1.2 × 10-8 and 4.6 × 10-8 m, respectively);16 and (c) threelayer micelles of M3 [0.5 wt % in D2O, radii of the core, poly(methyl methacrylate) (PMMA) layer and whole micelle 0.9 × 10-8, 1.2 × 10-8, and 4.8 × 10-8 m, respectively].10 These systems were chosen on the grounds of the following considerations: in system a no important diffusion barrier is probably offered by the interface as such (according to 1H NMR, the uptake of the solubilizate is almost instantaneous); in system b some polymer chain immobilization was detected at the interface, which could offer a slight diffusion barrier; and in system c CX enters preferentially the inner core made up by poly(2-ethylhexyl methacrylate) blocks, for which it is a good solvent, whereas the PMMA sheet rejects it and, thus, offers a substantial diffusion barrier.10 SDS Micelles. Each of the three solutes exhibits one sharp 1H NMR signal, which is slightly upfield shifted relative to that in the aqueous solution. In PGSE, all these signals have strictly monoexponential dependences on g2 as shown as an example in Figure 1 for CL at five different values of ∆. All three also have linear dependence of ζ (defined in eq 5) on ∆, suggesting, thus, unrestricted diffusion. As shown in Figure 2, these dependences have quite different slopes corresponding to different values of the apparent self-diffusion coefficients Dexp, which are given in Table 2. As shown in Figure 3, the signals of SDS micelles (decaying monoexponentially in PGSE) expectedly also have linear dependences of ζ on ∆. The corresponding self-diffusion coefficients Dm are also given in Table 2. It is notable that for CX, Dexp and Dm are equal within experimental error, which is less than 5 rel %. As expected from the linear dependence of ζ on ∆, the dependence of ζ on δ2 is linear, too, as illustrated in Figure 4. Thus, PGSE of all three solubilized diffusants has all signs of unrestricted diffusion, although the apparent (16) Krˇ´ızˇ, J.; Brus, J.; Plesˇtil, J.; Kurkova´, D.; Masarˇ, B.; Dybal, J.; Zune, C.; Je´roˆme, R. Macromolecules 2000, 33, 4108.
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Krˇ ı´zˇ
Figure 1. Logarithmic dependence of the relative signal intensity of CL on g2 in the PGSE experiment with δ ) 1 ms and ∆ as indicated (SDS/D2O solution, 300 K).
Figure 3. Dependence of ζ on ∆ in PGSE of SDS micelles with the indicated solubilized solutes (δ ) 1 ms, D2O, 300 K).
Figure 2. Dependence of ζ on ∆ in PGSE of the indicated solutes solubilized by SDS (δ ) 1 ms, D2O, 300 K).
Figure 4. Dependence of ζ on δ2 in PGSE of the indicated solutes solubilized by SDS (∆ ) 100 ms, D2O, 300 K).
Table 2. Partition Coefficients between Micelles and D2O and the Self-Diffusion Coefficients of the Micelles (Dm), Predicted (Dp) and Measured (Dexp) Values (PGSE, 300 K)
diffusion delay. Also, the measurements were done using the water-cooled high-resolution gradient probe in the gradient region 0-850 G/cm (instead of 0-50 G/cm used in the previous case). Despite the relative chemical shift between the signals of the micelle-absorbed and D2O-dissolved compound (∆δ ) -0.04, -0.06, and -0.05 ppm for CX, BZ, and CL, respectively), we observe only one signal of the solute in the 1H NMR spectra, which indicates a rather fast exchange between the micellar core and the aqueous environment (in the case of CX, the signal of D2O-dissolved compound is very weak so that it could avoid detection). The PGSE decays are strictly monoexponential again, and the dependences of ζ on ∆ are linear as in the case of SDS. The corresponding self-diffusion coefficients (with those of the M2 micelles) are given in Table 2. The linear dependence of ζ on ∆ even for CX, where the diffusion coefficient is equal to that of the micelle, again indicates unrestricted diffusion as characterized above. Three-Layer Micelles M3. In this case, the study is targeted on CX (for BZ and CL, M3 behaves in a quite analogous way as M2, which can be expected considering that the outer layer of its core is formed by the same polymer as the core of M2). In contrast to M2, CX can be only absorbed into the inner core, which is made up of the poly(2-ethylhexylacrylate) block.10 The outer core layer,
micellesa solute SDS SDS SDS M2 M2 M2 M3 a
CX BZ CL CX BZ CL CX
ξcs 5840 208 116 5242 297 80 3157
Dm Dp Dexp ×1011 m2 s-1 ×1011 m2 s-1 ×1011 m2 s-1 6.6 5.1 4.0 0.14 0.13 0.13 0.09
6.6 16.9 26.1 0.14 45.9 74.8 0.09
6.6 18.2 26.5 0.14 46.2 74.6 0.09
Concentration 3.0 wt % for SDS, 0.5 wt % in all other cases.
diffusion coefficients are very different from those of the same compounds dissolved in D2O. Interpretation of these observations will be given below. Two-Layer Micelles M2. In contrast to SDS, the highest concentration, under which micelles M2 and M3 give reliable results, is 0.5 wt % (used here). Above it, the micelles tend to form higher aggregates with a broad size distribution and correspondingly blurred solubilization and diffusion behavior. The self-diffusion of micelles and CX had to be measured using PGSTE (instead of PGSE) to avoid transverse relaxation of the spins during the
Molecule Diffusion in Semipenetrable Carriers
Figure 5. Logarithmic dependence of relative signal intensity of CX on g2 in the PGSTE experiment (δ ) 1 ms, values of ∆ indicated; M3/D2O solution, 300 K).
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size can be derived from the relation a ) (D∆)1/2 (cf. ref 15), which for D ≈ 10-10 m2 s-1 and ∆ ≈ 10-2 s amounts to 10-6 m, that is, micrometer size. Because our carriers are at least 2 orders of magnitude smaller, we can regard them as pointlike for PFG NMR. In the course of the PFG experiment, however, the spin vectors can de-phase due to three additional types of motion: (a) unrestricted selfdiffusion of the carrier (micelle) itself, characterized by the corresponding diffusion coefficient Dm; (b) occasional escape of a confined molecule and its diffusion in the surrounding medium; and (c) unrestricted self-diffusion of molecules dissolved in the medium, characterized by the diffusion coefficient Ds. An exact theoretical treatment of the de-phasing effects of all three motions together leads to rather cumbersome algebra, which is intractable in practical use. Therefore, we hazard an approximate approach. A statistically successful (no return) escape of a molecule into a hostile (low solubility) medium can be seen as overcoming two types of barriers: (R) the resistance of the interface ∆Ein and (β) the thermodynamic barrier given by the difference of the chemical potentials of the given substance in the core and the medium µc - µs. Because the dephasing probability must be given by the probability of escape times and the probability to move in the surroundings by diffusion, we can approximate the corresponding diffusion constant as
[
De ) exp
]
-∆Ein µc - µs Ds ) κinξcs-1Ds kT kT
(6)
where κin ) exp(-∆Ein/kT) is a permeability coefficient of the interface and the partition coefficient ξcs is
ξcs )
Figure 6. Dependence of ζ on 1/∆ in PGSTE of CX absorbed in the inner core of M3 micelles (δ ) 1 ms, D2O, 300 K).
made up of the PMMA block, can be swollen by CX to a very low degree only (CX is a precipitant for PMMA) and, thus, forms a barrier for its diffusion out of the core.10 In fact, 4 days of vigorous stirring of the M3 solution with CX at 330 K led to CX absorption, which is only near (but not at) equilibrium. Expecting the self-diffusion coefficient of CX inside the core to be at least 10-10 m2 s-1 and using δ ) 10-3 s and ∆ ) 0.1 s, both conditions ∆D . a2 and δD . a2 are fulfilled (a2 being10 equal to 0.81 × 10-16) so that the diffusion should be in the regime of rapid diffusion (eq 2). Accordingly, immeasurably low attenuations are expected even at the highest gradient (the exponents in eqs 2 and 3 for 850 G/cm are 3.1 × 10-16 and 8.4 × 10-13, respectively). This is contrary to the observations. Figure 5 shows measurable and strictly monoexponential decays for five various values of ∆. Figure 6 illustrates that the dependence of ζ on ∆ is linear again (experiments with BZ and CL are not included having incomparably higher values of ζ). In other words, the diffusion appears to be unrestricted. However, as shown in Table 2, the corresponding diffusion coefficient is equal to that of the whole micelle. Interpretation of Results. From the above discussion, it is clear that diffusion in nanoscaled spaces (a of the order of 10-8 m) is well below the resolution of a practically feasible PFG NMR. The practical lower limit of the probed
[
]
φc µc - µs ) exp 1 - φc kT
(7)
φc being the weight fraction of the given substance in the core. In equilibrium, the escape of molecules from the core is compensated by the entrance of other molecules into it. If De is very low and the signals of the substance in the core and in the medium are undistinguishable, we should obtain at least a biexponential decay in PFG (providing that the second signal is not incomparably weak). If De is large enough (which is the case of all cases presented here except CX), the exchange will merge not only the corresponding NMR signals into one but also the PFG responses; that is, the decay will follow the monoexponential relation
S(g) ≈ exp[-γ2g2δ2Dp∆]
(8)
where the predicted Dp is approximately
Dp ≈ φc[Dm + κinξcs-1Ds] + (1 - φc)Ds
(9)
The values of φc were obtained from the relative intensities of NMR signals of the given solute and the micelle, after correction for its known solubility in D2O. The calculated values of ξcs are given in Table 2 along with the predicted values of Dp (eq 9, κin was assumed to be 1.0 in all cases except M3, where it was put to 0). Comparison of the predicted values Dp with those obtained experimentally, Dexp, shows that the approximation represented by eqs 8 and 9 works rather well in the present cases. In view of the underlying considerations, it should work equally well for similar systems.
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Conclusions According to the results of the present study, PFG NMR cannot see restricted diffusion of small molecules confined in mobile nanoscale carriers directly. The reason is that the small size of the restricted space does not allow measurable spin dephasing at practical gradient values. However, it can probe partial restriction of diffusion indirectly by analyzing the PFG decay obtained under suitable conditions. The analysis includes measuring (1) the equilibrium partitioning of the given substance between the carrier and the surrounding medium; (2) selfdiffusion of the substance in the medium; and (3) selfdiffusion of the carrier as a whole. From these quantities and the experimentally obtained diffusion constant Dexp, one can extract the permeability coefficient κ characterizing the structural barrier of penetrating the interface by the molecules under study. In the present study probing the diffusion of CX, BZ, and CL absorbed in the D2O solutions of (a) monolayer
Krˇ ı´zˇ
micelles of SDS, (b) two-layer micelles of M2, and (c) threelayer micelles of M3, the barrier hindering the diffusion out of the micelle appears to be the thermodynamic one given by the solubility in the medium. In the case of CX and M3 (case 3), additional hindering by a structural barrier can be inferred from other experiments but it cannot be demonstrated by PFG NMR because of the low solubility of CX in water. Acknowledgment. The author thanks Prof. Eiichi Fukushima for the discussion of some problems of PGF NMR in restricted space. He also thanks the Grant Agency of the Academy of Sciences of the Czech Republic for financial support given under Grant A4050206 and the Academy of Sciences of the Czech Republic for additional support (Project Nos. AVOZ4050913 and KSK4050111). LA048690Y
