**3. Surface Grating Decay and Surface Diffusion**

Surface mobility can considerably affect processes including nucleation, crystal growth, catalysis, and sintering. The high mobility of surface molecules originates from the special coordination environment, where molecules have fewer neighbors and a greater degree of freedom compared to the molecules in the interior [20,38,39]. For amorphous pharmaceutical solids, a high surface mobility causes the rapid nucleation and crystal growth at the free surface or interior interface, largely determining the physical stability [20,25,27,40]. Moreover, a high surface mobility could also allow the efficient equilibrium of newly deposited molecules in vapor deposition, facilitating the formation of highly stable glass with exceptionally low energy and high density [41,42].

Surface grating decay method has been widely used to measure the surface diffusion of pure amorphous drug and polymer-doped solid dispersion in the pharmaceutical field [20,43–45]. Herein, surface diffusion represents the lateral movement of molecules at the free surface. As shown in Figure 2, a master grating with gold coating is placed on the surface of drug liquid below *T*g to print the surface grating. The master can be detached at a lower temperature and a pharmaceutical glass with a corrugated surface is produced. The smoothing of the surface grating is followed with an atomic force microscope or optical microscope under nitrogen atmosphere. The amplitude of surface grating could be obtained by the Fourier transformation of the height profile of each scan line. In general, grating amplitude *h* decreases exponentially following *h* = *h*<sup>0</sup> exp(−*Kt*) in the supercooled

liquid state. For comparison, *h* decays slightly deviating from exponentially and could be described as *<sup>h</sup>* <sup>=</sup> *<sup>h</sup>*<sup>0</sup> exp[−(*Kt*) *<sup>β</sup>*] in the glassy state, one phenomenon mainly attributed to the glass aging. Here, K represents the decay constant and *β* is slightly smaller than 1. According to the Mullin's model, the decay constant K can be described as a combination of individual processes including viscous flow (F), evaporation condensation (A and A'), bulk diffusion (C), and surface diffusion (B) in the following equation.

$$\mathbf{K} = \mathbf{F}q + \mathbf{A}q^2 + Dq^3 + Bq^4$$

$$\mathbf{q} = \frac{2\pi}{\lambda} \quad \mathbf{F} = \frac{\gamma}{2\eta} \quad \mathbf{A} = \frac{p\_0 \gamma \Omega^2}{(2\pi m)^{0.5}(kT)^{1.5}} \tag{1}$$

$$\mathbf{D} = \mathbf{A}' + \mathbf{C} = \frac{\rho\_0 D\_G \gamma \Omega^2}{kT} + \frac{D\_v \gamma \Omega}{kT} \quad \mathbf{B} = \frac{D\_s \gamma \Omega^2 v}{kT} . \tag{2}$$

where *γ* represents the surface free energy, *η* represents the viscosity, *p*<sup>0</sup> represents the equilibrium vapor pressure, Ω represents the molecular volume, *m* represents the molecular mass, *ρ*<sup>0</sup> represents the equilibrium vapor density, *DG* represents the diffusion coefficient of evaporated molecules, and *v* represents the number of molecules per unit area of surface. *Dv* and *Ds* is the coefficient of bulk diffusion and surface diffusion, respectively.

**Figure 2.** The experimental scheme of surface grating decay for surface diffusion measurement. Adapted from [43] with the permission. Copyright © 2020 American Chemical Society.

In 2011, Zhu et al. investigated the surface diffusion of a small-molecular drug indomethacin by the method of surface grating decay for the first time [44]. Surface evolution of amorphous indomethacin is mainly controlled by the viscous flow at high temperature while the mechanism of surface evolution changes to the surface diffusion with the temperature decreasing to near and below *T*<sup>g</sup> [20,43–45]. Compared to the bulk diffusion, surface diffusion of amorphous drugs can be orders of magnitude faster [44–46]. Moreover, as shown in Figure 3, surface diffusion coefficient *D*<sup>s</sup> has been shown to be roughly proportional to the velocity of surface crystal growth *u*s (*u*s~*D*s 0.87), indicating the controlling role of *D*s in the process of surface crystal growth [45].

Recent studies showed that surface diffusion of molecular glasses can be strongly affected by several factors, including strength of molecular interaction [47,48], molecular size [49], and the addition of polymer [43]. With the increase in the strength of molecular interaction and molecular size, surface diffusion of amorphous solids exhibits a tendency to slow down [47–51]. For instance, Chen et al. found that the surface diffusion of polyalcohol glasses showing extensive hydrogen bonding is much slower than that of the molecular glasses of comparable size but with no or limited hydrogen bonds [48]. They proposed that the inhibition of surface diffusion in systems containing extensive hydrogen bonding interactions is mainly attributed to the abundance of hydrogen bonds near the surface [48]. As a result, the loss of nearest neighbors could not induce a proportional decrease in the kinetic barrier of surface diffusion [48]. Surface diffusions of posaconazole and itraconazole were also investigated by surface grating decay method, and the diffusion rates of these two rod-like molecules are much lower than those of quasi-spherical molecules of similar

volume, a result of the deep penetration of rod-like molecules in the bulk where molecular mobility is slow [52]. In amorphous systems without extensive hydrogen bonds, surface diffusion coefficients of molecular glasses were proposed to decrease with an increase in their penetration depth [52]. In addition, Mokshin et al. proposed that surface diffusion coefficient *D*s is directly related to the kinetic coefficient (also termed as attachment coefficient) of crystallized molecular glasses [53]. In a very recent study, Bannow et al. investigated the effects of a commercial polymeric excipient Soluplus on the surface diffusion of amorphous indomethacin [43]. The addition of low-concentration Soluplus significantly slowed down the surface diffusion of indomethacin [43]. Further increase in the concentration of Soluplus would lead to turnover, where the increasing inhibitory effect of Soluplus on the surface diffusion with the Soluplus concentration increasing becomes less pronounced [43]. Moreover, the decrease in surface diffusion of amorphous indomethacin by doping low content Soluplus correlates well with the enhanced physical stability [43].

**Figure 3.** Surface crystal growth rate *u*s, plotted against surface diffusion coefficients *D*s for reported glasses and silicon. Adapted from [45] with the permission. Copyright © 2017 American Chemical Society.
