Diffusion Measurements of Water and Polymers in Hydrogels by Pulsed Field Gradient NMR


Samples

Temperature/°C

Squeezing

Setting

Inlet

Column

(a) Prepared by dissolving agar powder in water

–

95

60

60

(b) Squeezed from 2.3 wt% agar gel

25

r.t.

30

30

(c) Squeezed from 2.3 wt% agar gel and heated at 95 °C for 1 h

25

95

60

60



The tumbling and translational motions of polysaccharides change markedly during aggregation and gelation. NMR measurements provide information on the mobility of molecules in gel. The spin-lattice relaxation time (T 1) and the spin-spin relaxation time (T 2) reflect the local mobility and flexibility of the molecules, respectively, and D measured by field gradient NMR reflects the displacement of a molecule by self-diffusion.

In this chapter, we introduce the basic theory and techniques of diffusion measurements by pulsed field gradient NMR in Sect. 6.2 and experimental results for solutions and gels of poly(N,N-dimethylacrylamide) [4], carrageenans [8, 9], agar [10], and agarose [11] in Sect. 6.3.



6.2 Diffusion Coefficient Measurements


An application of field gradient attaches a spatial information to the NMR signal. Therefore, the gradient can produce a spatial distribution of nuclei, that is, NMR imaging. When two field gradients for the diphase and rephase are applied, the NMR signal decays due to the displacement of nuclei during the interval between the two field gradients. This allows calculation of the diffusion coefficient for Fickian diffusion in free space. In this section, the use of field gradient for diffusion measurements is described.

The Larmor precession frequency depends on the magnetic field experienced by the nucleus and is therefore dependent on nuclei position when the field gradient is present. The spatially dependent Larmor frequency ω (r) at the position r under a spatially linear field gradient g is expressed as follows:


$$ \omega \left(\mathbf{r}\right)=\gamma \left({H}_0\mathbf{g}\;\mathbf{r}\right)={\omega}_0+\gamma\ \mathbf{g}\;\mathbf{r} $$

(6.1)
where H 0 is the externally applied magnetic field and g = 0 at r = 0. It follows that the phase angle ϕ is a function of the spatial position of the nucleus. If the field gradient duration is δ, then ϕ is given with


$$ \phi (r)=\upgamma\;\mathbf{g}\;\mathbf{r}\;\delta $$

(6.2)
The distance in the direction of g where ϕ(r) = 2 π is


$$ {q}^{-1}=2\uppi /\upgamma\ \mathrm{g}\;\delta $$

(6.3)
q −1 is the characteristic length scale of the field gradient. For example, q −1 = 235 μm for g = 10 G/cm with δ = 1 ms. When the sample size, or the size of detection area, is several times larger than q −1, the total signal intensity vanishes because of dephasing.

For diffusion coefficient measurements, a second field gradient is applied in order to rephase the dephased magnetization. Figure 6.1 shows a typical pulse sequence with two rectangular-shaped pulsed field gradients (PFG) along the z axis [12] and the dephasing and rephasing behavior of the magnetization when individual spins do not change positions in the interval Δ between the two PFGs. (a) The magnetizations are aligned along the y axis by an r.f. π/2 pulse. (b) Under the first PFG, the magnetizations precess at an angular velocity of γ g r corresponding to the individual positions in the rotating frame. (c) At the end of the first PFG, the magnetizations are spirally twisted at a pitch of q −1. (d) The application of an r.f. π pulse along the y axis rotates the individual magnetizations 180° about the direction of y axis, which yields a mirror-symmetrical arrangement of the magnetizations with respect to the y–z plane. (e) Under the second PFG, the individual magnetizations precess at the same angular velocity with that under the first PFG. (f) At the end of the second PFG, the magnetizations are aligned along the y axis. When the nucleus is displaced Δz in the z direction during Δ, its phase angle is

A314557_1_En_6_Fig1_HTML.gif


Fig. 6.1
A typical pulse sequence with two pulsed field gradients of rectangular shape and the dephasing and rephasing behavior of the magnetization. (a) The magnetizations are aligned along the y axis by an r.f. π/2 pulse. (b) Under the first PFG, the magnetizations precess at the angular velocity of γ gr corresponding to the z coordinate. (c) At the end of the first PFG, the magnetizations are spirally twisted at a pitch of q −1. (d) An r.f. π pulse along the y axis rotates the individual magnetizations along the direction of y axis through 180°. (e) Under the second PFG, the individual magnetizations precess at the same angular velocity with that under the first PFG. (f) At the end of the second PFG, the magnetizations are aligned along the y axis



$$ \varphi \left(\varDelta z\right)=2\pi \frac{\Delta z}{q^{-1}}=\gamma g\delta \kern0.1em \Delta z $$

(6.4)
The echo signal intensity I(2τ,g δ) at 2τ is proportional to the vector sum of magnetizations in the sample, therefore expressed as follows:


$$ \mathrm{I}\left(2\tau, g\delta \right)=\mathrm{I}\left(2\tau, 0\right){\displaystyle \int \cos \left(\phi \left(\Delta z\right)\right)\rho \left(\mathbf{r}\right)p\left(\mathbf{r},\Delta z\right)d\mathbf{r}d\Delta z} $$

(6.5)

where ρ(r) is the density of the nucleus and is constant for homogeneous sample, p(r, Δz) is the probability of the displacement during Δ for the nucleus at r, and I(2τ,0) is the total signal intensity without PFG and expressed as follows:


$$ I\left(2\tau, 0\right)=I\left(0,0\right) \exp \left(-2\tau /{T}_2\right) $$

(6.6)
where I(0,0) is the initial signal intensity just after the r.f. π/2 pulse. For free diffusion in an isotropic medium, p(r, Δz) becomes a Gaussian distribution


$$ p\left(r,\Delta z\right)={\left(4\pi D\Delta \right)}^{-1/2} \exp \left(-\frac{\Delta {z}^2}{4D\Delta}\right) $$

(6.7)
where D is the diffusion coefficient. Taking the diffusion during δ into account, I(2 τ,gδ) is rewritten as follows:


$$ I\left(2\tau, g\delta \right)=I\left(0,0\right) \exp \left(-\frac{2\tau }{{\mathrm{T}}_2}-{\left(\gamma \mathrm{g}\delta \right)}^2D\left(\Delta -\frac{\delta }{3}\right)\right) $$

(6.8)
In common measurements of the free diffusion, gδ is varied under constant Δ.

In order to decrease the effect of residual field gradient, the pulse sequence of pulsed gradient stimulated spin echo (PGSTE) is frequently used for D measurements [13–17]. The diffusion coefficient values are determined from the decay of echo signal intensities, expressed as follows:


$$ I\left(2{\tau}_2+{\tau}_1g\delta \right)=I\left(2{\tau}_2+{\tau}_1,0\right) \exp \left[-{\gamma}^2{\delta}^2{g}^2\left(\varDelta -\delta /3\right)D\right] $$

(6.9)
where I(2τ 2 + τ 1, gδ) and I(2τ 2 + τ 1,0) are echo signal intensities at t = 2τ 2 + τ 1 with and without the field gradient pulse, respectively, γ is the gyromagnetic ratio of the measured nucleus, g is the field gradient strength, δ is the duration of the gradient pulse and Δ is the diffusion time, equal to the interval between two field gradient pulses. Note that I(2τ 2 + τ 1, 0) has decayed from the initial intensity, I(0, 0), (i.e., the signal intensity immediately after the first π/2 rf pulse) by T 1 and T 2 relaxations:


$$ I\left(2{\tau}_2+{\uptau}_1,0\right)=I\left(0,0\right) \exp \left[-2{\uptau}_2/{T}_2-{\uptau}_1/{T}_1\right] $$

(6.10)
From Eq. 6.10 it follows that a decrease in the relaxation times leads to a decrease in I(2τ 2 + τ 1,0).


6.3 Diffusion of Water and Polymers in Hydrogels



6.3.1 Molecular Diffusion in Poly(Dimethylacrylamide) Gel


Most synthetic polymer gels have chemically cross-linked network structures. As a result the polymer concentration is usually homogeneous above a length scale of several nm. Conversely, physically cross-linked gels such as polysaccharide gels, where aggregations are responsible for gelation, have concentration fluctuations over larger length scales.

In this section we look at the diffusion of water and a probe polymer, polyethylene glycol (PEG), in a gel of poly(dimethylacrylamide) (PDMAA), a synthetic polymer.


6.3.1.1 Diffusion of HDO in PDMAA


The diffusion coefficient of HDO (D HDO) in a PDMAA gel was determined by the PGSE 1H NMR method at 303 K varying the swelling ratio Q, which is inversely proportional to the concentration of the network polymer, and the Mw of PEG added as a probe. As seen from Fig. 6.2, D HDO increases as Q is increased but is almost independent of the Mw of PEG. The Q-dependence of D HDO is strong at low Q values and is much more moderate at high Q values. The D HDO value in the large-Q region asymptotically approaches that for HDO in neat D2O (2.22 × 10−9 m2s−1). We can conclude that intermolecular interactions between water and the polymer network restrict the translational motion of water. We also see that the diffusion of HDO is independent of the Mw of PEG. We can understand more clearly the dynamical behavior of water in the gel system by appealing to the following modified free volume theory equation:

A314557_1_En_6_Fig2_HTML.gif


Fig. 6.2
Dependence of the diffusion coefficient of water molecule (D HDO) on the degree of swelling (Q) in a PDMAA gel containing PEG with Mw = 4,250 (□), 10,890 (▲), and 20,000 (O) at 303 K



$$ {D}_{\mathrm{HDO}}={D}_{\mathrm{HDO},0}\kern0.5em \exp \left[{Q}^{-1}/\left({Q}^{-1}{f}_{\mathrm{solv}}-{f}_{\mathrm{solv}}^2/\beta \right)\right] $$

(6.11)
where f solv, β, and D HDO,0 are the fractional free volume of deuterated water, the proportionality constant in the free volume theory, and D HDO in neat solvent, respectively. The solid curve obtained from a least-squares fitting of the experimental data to Eq. 6.11 is shown in Fig. 6.2. The theoretical fit agrees well with the experimental data. From this result, the diffusion coefficient of HDO in the neat solvent (Q → infinity) is D HDO,0 = 2.16 × 10−9 m2s−1, which is lower than HDO in neat D2O (2.22 x 10−9 m2s−1). This means that the diffusion coefficient for HDO in neat D2O is different from the Q-extrapolated value to the infinite Q given by the modified free volume theory for the Q-dependence of the diffusion coefficient. The diffusion coefficient for HDO in bulk PDMAA (Q → 1) obtained from Fig. 6.2 was 2.5 × 10−11 m2s−1. Clearly, the translational motion of HDO in the limit Q → 1 (bulk polymer) is restricted compared to that in gels. Further, the rapid decrease of D HDO in the small-Q region can be explained by a decrease in free volume.


6.3.1.2 Diffusion of PEG in PDMAA


Assuming that diffusion of a hard-sphere probe molecule is restricted mainly by hydrodynamic interactions with the host polymer, its mobility can be related to its radius and the screening length of the polymer network. A mean field treatment of the hydrodynamic model yields


$$ D/{D}_0= \exp \left(-\kappa R\right) $$

(6.12)
where D is the diffusion coefficient of the probe in the host solution or gel, D 0 is the diffusion coefficient of the probe polymer in the neat solvent, R the radius of the spherical probe, and κ −1 the hydrodynamic screening length of polymer chain. When κ −1 is equivalent to R, D/D 0 = e −1 ≈ 0.37. This approach has been used to analyze the diffusion of probe molecules in semidilute polymer solutions [2] and polymer gels [3]. It has also been employed to analyze the self-diffusion of random coil polymers in solutions [18] and gels [19], in which κ −1 is large enough that the diffusion of random coil polymers is controlled by their hydrodynamic radius R H, as opposed to highly restricted reptation, where besides R H internal dynamics play a part. When κ −1 is large enough, the hydrodynamic interactions are more important than topological constraints, and R in Eq. 6.12 can be replaced with R H. When the topological constraints cannot be neglected, compared with the interchain hydrodynamic interactions, D becomes larger than the value expected from Eq. 6.12. When the diffusion coefficient of an isolated PEG (D PEG,0) is determined from the diffusion coefficient of PEG in dilute aqueous solution(D PEGsoln), it is necessary to take into account the change of local friction of polymer, which is estimated from the change of the diffusion coefficient of the solvent [20]. Then we have


$$ {D}_{\mathrm{PEG},0}={D}_{\mathrm{PEG},\mathrm{soln}}/\left({D}_{\mathrm{HDO},\mathrm{neat}}/{D}_{\mathrm{HDO}}\right) $$

(6.13)
where D HDO,neat is the diffusion coefficient of HDO in D2O.

The diffusion coefficient of PEG (D PEG) of PEG in PDMAA gels was determined by the PGSE 1H NMR method at 303 K for different Q values and Mw values of PEG. The D PEG values are plotted against Q in Fig. 6.5a. It was found that D PEG increases as Q is increased and also depends on the Mw of PEG. The D PEG value in the gel is smaller than in a 1 wt% aqueous PEG solution (D PEG,soln), as shown in Fig. 6.5b. We can conclude that the translational motion of PEG in the gel, compared to that in dilute solution, is restricted through intermolecular interaction with the polymer network. This restriction becomes stronger with increasing Mw of the probe. The diffusion of PEG in the gel can be analyzed using Eq. 6.12. The relationship between the hydrodynamic screening length (k −1) and the concentration of network polymer (c) or the degree of swelling (Q) is expressed by


$$ {k}^{-1}\kern0.2em \sim \kern0.2em {c}^u={Q}^{-u} $$

(6.14)
where −u is in the range 0.5–1.0 for different polymers [1]. Substitution of Eqs 6.13 and 6.14 into Eq. 6.12 gives


$$ \ln \left(- \ln \left({D}_{\mathrm{PEG}}/{D}_{\mathrm{PEG},0}\right)\right)=u \ln Q+ \ln \kern0.2em {R}_{\mathrm{H}} $$

(6.15)
where


$$ {R}_{\mathrm{H}}=\mathrm{k}T/6\uppi \eta \mathrm{s}\;{D}_{\mathrm{PEG}} $$

(6.16)
Using Eq. 6.16, Eq. 6.15 is rewritten as follows:


$$ \ln \left(- \ln \left({D}_{\mathrm{PEG}}/{D}_{\mathrm{PEG},0}\right)\right)=\mathrm{u} \ln Q+ \ln \left(\mathrm{k}T/6\uppi \eta \mathrm{s}\right) $$

(6.17)
In Fig. 6.3, the ln[(−ln(D PEG/D PEG,0)) D PEG,soln] is plotted against ln Q. A straight line with a slope of −0.71 is obtained, which yields u = −0.71. Therefore, we have

A314557_1_En_6_Fig3_HTML.gif


Fig. 6.3
Plot of ln[(−ln(D PEG/D PEG,0) D PEG,soln] against ln Q at 303 K. M ws of PEGs used are 4,250 (□), 10,890 (▲), and 20,000 (O)



$$ {k}^{-1}\kern0.2em \sim \kern0.2em {Q}^{0.71}={c}^{-0.71} $$

(6.18)
Several theoretical treatments have been proposed to predict the scaling exponent u, most of them taking into account the network structure and interaction with the solvent [1]. de Gennes [21] proposed the relationship k −1 ~ c -0.75 for flexible polymer chains diffusing in a host polymer network in a good solvent. The experimental results for the random coil PEG in a PDMAA are close to de Gennes’ prediction.


6.3.2 Gelation Mechanism and Molecular Diffusion in Carrageenan Gels


κ-Carrageenan comes from a family of linear water-soluble polysaccharides extracted from different species of marine red algae. The chemical structure of κ-carrageenan is characterized by an alternating disaccharide composed of α-1, 3-linked galactose-4-sulfate and β-1, 4-linked 3, 6-anhydrogalactose (Fig. 6.4). It is largely used in the food [22], pharmaceutical [23, 24], and cosmetic [25] industries as a gelling and thickening agent, texture enhancer, or stabilizer. The gelling properties of κ-carrageenan have been extensively studied by many researchers [25–31]. The gelling process is generally accepted as a two-step model involving a coil-to-helix conformational transition followed by aggregation of the ordered molecules to form an infinite network [32, 33]. This aggregation leads to a decrease in the solute (random coil) concentration of κ-carrageenan, and therefore, molecular diffusion of probe molecules should be enhanced. This diffusion behavior can provide microscopic insights into the gelation mechanism of κ-carrageenan.

A314557_1_En_6_Fig4_HTML.gif


Fig. 6.4
Chemical structure of κ-carrageenan


A314557_1_En_6_Fig5_HTML.gif


Fig. 6.5
(a) Stacked PGSTE 1H spectra of a 0.1 % solution of pullulan with M w = 10.7 × 104 g/mol at 25 °C as a function of the field gradient strength g (g was varied from 1.8 to 6.0 T/m in the vertical direction). The residual water peak is indicated by a “w”; (b) Stacked PGSTE 1H spectra of a 1 % κ-carrageenan solution containing 0.1 % pullulan with M w = 10.7 × 104 g/mol as a probe polymer during the cooling process. The peaks selected for the diffusion analysis of the pullulan and κ-carrageenan are indicated by “p” and “κ,” respectively. The field gradient strength, g, was varied from 2.0 to 7.0 T/m in the vertical direction

Recently, the diffusion of probe molecules in κ-carrageenan gels has attracted much attention since it is closely related to a variety of applications, such as release of drugs from gels and release of flavor compounds encapsulated in foods [34–37].

In this section we describe a method to estimate hydrodynamic mesh size of k-carrageenan solutions and gels from diffusion measurements of pullulan, added as a probe. The flexibility of the carrageenan chain during aggregation and gelation is also discussed on the basis of 1H T 2 measurements.


6.3.2.1 Pulsed Field Gradient 1H Spectra of Pullulan


Figure 6.5a shows the stacked 1H spectra of a 0.1 % pullulan aqueous solution. The signal intensity of pullulan decayed with increasing g, reflecting the diffusion of pullulan.

Figure 6.5b shows the pulsed field gradient 1H spectra of a 1 % κ-carrageenan solution containing 0.1 % pullulan at various temperatures. The peaks at 5.36 and 5.17 ppm, which are assigned to the anomeric proton of pullulan and the C1 proton on the β-1, 4-linked 3, 6-anhydrogalactose unit of κ-carrageenan, respectively, were chosen for the analysis. The corresponding signals are indicated by p and κ, respectively. The water peak at 4.7 ppm was completely eliminated due to the fast diffusion of water.

The intensity of the peak from the anomeric proton of pullulan at 5.36 ppm, which is separated from the peaks of κ-carrageenan, decayed with increasing g and indicates the diffusion of pullulan in the gel structure formed by κ-carrageenan.

Based on Eq. 6.9, a semilogarithmic plot of I pull(g) as a function of γ 2 δ 2 g 2(Δ-δ/3) gives a straight line with a slope of −D if the diffusant has a single diffusion process. I pull (g) values at 25 °C in D2O and in κ-carrageenan gels, obtained from the data Fig. 6.5, are represented in semilogarithmic plots in Fig. 6.3 and lie on a straight line, indicating that pullulan undergoes single-mode diffusion in both cases. The diffusion coefficient of pullulan, D pull, can be similarly obtained. The diffusion coefficient of pullulan in the κ-carrageenan solution is smaller than that in neat D2O. This means that the diffusion of pullulan is restricted by the κ-carrageenan network. This effect [38, 39] will be discussed in detail later.


6.3.2.2 Diffusion of Pullulan in Neat Solvent


D pull values in 0.1 % solution of pullulan (D pull,0) were measured for pullulan with various M w at 25 °C and plotted against M w in Fig. 6.6. If the diffusion follows Zimm dynamics, that is, there is no draining of solvent molecules from the interior of the polymer, then, the relationship between M w and the diffusion coefficient in a dilute solution, D 0, can be expressed as [40]

A314557_1_En_6_Fig6_HTML.gif


Fig. 6.6
Double-logarithmic plot of the diffusion coefficient of pullulan in D2O, D pull,0 as a function of the molecular weight M w of pullulan at 25 °C (●). The line is a fit of equation of D 0 = a M w ν to the data



$$ {D}_0=a\;{M}_{\mathrm{w}}^{-\upnu}. $$

(6.19)
where a is a constant related to the segment size of the polymer chain and ν is a scaling exponent that depends on the polymer-solvent system. As seen in Fig. 6.6, the double-logarithmic plot of D pull,0 as a function of M w is essentially linear, and the experimental result gives the following equation:
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Mar 22, 2018 | Posted by in BIOCHEMISTRY | Comments Off on Diffusion Measurements of Water and Polymers in Hydrogels by Pulsed Field Gradient NMR

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