Normal breathing (tidal breathing) is an unconscious activity. In fact, if one is told to "breathe normally," one usually is unable to do so because breathing has moved to a more conscious level. When we breathe for life purposes, inhalation is active and exhalation is passive; in other words, we use muscles to get air into the lungs but air is forced out by the natural recoil of the respiratory system. The primary muscle of life breathing is the diaphragm, but it important to note that this muscle is active during inhalation, but not exhalation.
Speech breathing is a more conscious activity than life breathing and requires more muscular effort. While inhalation and exhalation are roughly of equal duration during tidal breathing (about 40% of a respiratory cycle is spent inhaling, 60% exhaling), we need to get air in quickly and exhale it gradually for speech, resulting in about a 10%-90% inspiratory-expiratory ratio per cycle.
The diaphragm and external intercostal muscles (located between the ribs) are typically active during speech inhalation and, initially, the external intercostals are active during the beginning of exhalation for speech as they "check" the nonmuscular forces from the lungs. After a certain point, the internal intercostal muscles activate and assist those passive forces and, depending on how much we have to say, other (accessory) muscles may come into play, particularly at lower lung volumes.
However, the diaphragm is not active as an exhalatory muscle for speech. The reason for this difference in muscle activity in speech breathing vs. life breathing is that our goal during speech is to maintain a constant pressure (subglottal pressure) to drive the speech mechanism and maintain a steady voice.
Although the diaphragm is said to contain two muscles or two portions - the crural and costal muscles (Pickering & Jones, 2002), it does not appear that these muscles or portions of the diaphragm can be activated or controlled independently (see quote from Pickering below). Readers can learn more about breathing and speech breathing from any basic speech science textbook (e.g., Hixon, Weismer, & Hoit, 2008; Raphael, Borden, & Harris, 2011).