# Shape Memory Effect in Micro-Sized Shape Memory Polymer Composite Chains
## 6 Shape Recovery Properties of Shape Memory Polymer/Nickel (SMP/Ni) Vertical Chain Arrays
In  
Figure 10: Typical shape recovery cycle of the vertical protrusive Shape Memory Polymer/Nickel (SMP/Ni) chains (2 vol% Ni). (**a**) Original shape; (**b**) Flattened pre-deformed shape; (**c**) Recovered shape. Figure 9: Typical configurations of flattened Shape Memory Polymer/Nickel (SMP/Ni) chain array with 2% nickel particles. (**a**) Morphology of flattened pillars; (**b**) Zoomed-in view of (a). local nonuniform magnetic field, they rotate, repel or attract between two adjacent nickel particles depending on the direction of induced dipole moments and their relative position [33; 34; 35; 36]. Referenced to the aforementioned studies of magnetorheological fluids, the formation of nickel chain structures in SMP matrix should consider the following forces applied on nickel particles:  
* Magnetic force. SMPs are macromolecule and not ferromagnetic materials, and their magnetic susceptibility is negligible. The magnetic forces \(\mathbf{F}_{j}^{\text{mag}}\) acting on a single particle \(j\) are induced by external magnetic field as well as induction field among nickel particles themselves [35]: \(\mathbf{F}_{j}^{\text{mag}}=\mathbf{m}\cdot\
abla(\mathbf{B}_{0}+\sum\limits _{i\
eq j}\mathbf{B}_{i})\), where \(\mathbf{m}\) indicates dipole moment, \(\mathbf{B}_{0}\) reveals magnetic induction intensity of external magnetic field, \(\mathbf{B}_{i}\) indicates magnetic induction intensity of particle \(j\) induced by particle \(i\). * Repelling force. Repelling forces exist when a nickel particle \(j\) approaches other particles (\(\mathbf{F}_{j}^{\text{sphere}}\)). Moreover, repulsive forces of a nickel particle \(j\) from walls of container (\(\mathbf{F}_{j}^{\text{wall}}\)) are included. * Gravitation force. \(\mathbf{F}_{j}^{\text{gav}}\). The gravitation of a nickel particle \(j\). * Hydrodynamic force. The hydrodynamic force \(\mathbf{F}_{j}^{\text{hydro}}\) of a moving nickel particle \(j\) for small Reynolds can be descripted by [PERSON]'s equation \(\mathbf{F}_{j}^{\text{hydro}}=-D\frac{d\mathbf{r}_{j}}{dt}\), where \(D=6\pi a\eta\) reveals Stokes drag coefficient of a nickel sphere. \(\eta\) is the viscosity coefficient of the SMP/DMF solution. Note that, the viscosity coefficient increases during solidification process. Hence, on the basis of references [33; 34], the motion governing equations of a nickel particle \(j\) in the SMP/DMF solution can be expressed as  
\[m\frac{d^{2}\mathbf{r}_{i}}{dt^{2}}+D\frac{d\mathbf{r}_{i}}{dt}=\mathbf{F}_{i }^{\text{mag}}+\mathbf{F}_{i}^{\text{sphere}}+\mathbf{F}_{i}^{\text{wall}}+ \mathbf{F}_{i}^{\text{gav}}. \tag{1}\]  
The properties of vertical protrusive chains (diameter, height, and interval between two adjacent chains) can be explained as follows according to Equation (1). For finitely long chains, the interaction between two adjacent chains may be attraction or repelling forces depending on their interval, and height of the two chains. Those interaction forces cause the local gather of adjacent nickel chains in local nonuniform magnetic field; hence, big and high nickel chains atop SMP substrate form. Naturally, with more nickel particle fillers, the diameter and height of nickel chains will be larger, and the interval will be smaller. After solidification of SMP/DMF/Ni mixture, the SMP/Ni chains form with different diameter, height, and interval. On the other hand, the forces among nickel chains are in short range, and the stable protrusive array forms. The field between neighboring protrusive chains is relatively stable and almost independent on the exact nickel content. Because of the complexity and imperfection of the above theoretical models referenced to magnetomechanical coupling theory of the magnetorheological fluids [26; 33; 34; 35; 36], it is still difficult to quantitatively design the distributions or accurately control the dimensions (diameter, height, and interval distance). Alternatively, in order to form well distributions of SMP/Ni chain arrays as much as possible, the experimental methods are still employed.